Conic Section
151 JEE Maths previous year questions on Conic Section — options free on every question; 15 include the answer & explanation free, the rest unlock with PYQ Pass.
Let \(P\)be the parabola, whose focus is \((-2,1)\) and directrix is \(2x+y+2=0\). Then the sum of the ordinates of the points on \(P\), whose abscissa is \(-2\) , is
[JEE Main 2025, 7 Apr (Shift 1)]
\(\frac{3}{2}\)
Let point \(P(-2,y)\) is point on parabola from def. of parabola
\(PS=PM\)
\(\Rightarrow {(y−1)}^{2}=\frac{{(−4+y+2)}^{2}}{5}\)
\(\Rightarrow 5{(y−1)}^{2}={(y−2)}^{2}\)
\(\Rightarrow 4{y}^{2}−6y+1=0\)
Sum of roots \(=\frac{6}{4}=\frac{3}{2}\)
A rod of length 8 units having two end points always lie on \(x-y+2=0\) and \(x+y+2=0\). A point \(P\) divide this line in ratio \(2: 1\). Then locus of \(P\) is
\(9 x^2+9 y^2+36 x-28=0\)
\(\begin{aligned}& x-y+2=0 \\& x+y+2=0\end{aligned}\)
\( P Q=8 \)
\(\Rightarrow(a-b)^2+(a+2+b+2)^2=64 \)
\((a-b)^2+(a+b+4)=64\)
\(2 a^2+2 b^2-2 a b+16+2(a b+4 b+4 a)=64 \)
\(\Rightarrow 2 a^2+2 b^2+8 a+8 b=48 \)
\(\Rightarrow a^2+b^2+4 a+4 b-24=0\)
\(\Rightarrow (a+2)^2+(b+2)^2=32\)
$$\begin{aligned}& h=\frac{2 b+1(a)}{3}, k=\frac{2(-2-b)+1(a+2)}{3} \\& 3 h=2 b+a, 3 k=a-2-2 b \rightarrow \text { Solve for } a \text { and } b . \\& a=\frac{3 h+3 k+2}{2}, b=\frac{3 h-3 k-2}{4} \\& (a+2)=\left(\frac{3 h+3 k+6}{2}\right), b+2=\left(\frac{3 h-3 k+6}{2}\right) \\& \Rightarrow\left(\frac{3 h+3 k+6}{2}\right)^2+\left(\frac{3 h-3 k+6}{2}\right)^2=32 \\& 9(x+y+2)^2+9(x-y+2)^2=128 \\& 18\left[x^2+y^2+4 x+4\right]=128 \Rightarrow x^2+y^2+4 x-\frac{28}{9}=0\end{aligned}$$\)
Let \(H: \frac{-x^2}{a^2}+\frac{y^2}{b^2}=1\) be the hyperbola, whose eccentricity is \(\sqrt{3}\) and the length of the latus rectum is \(4 \sqrt{3}\). Suppose the point \((\alpha, 6), \alpha>0\) lies on \(H\). If \(\beta\) is the product of the focal distances of the point \((\alpha, 6)\), then \(\alpha^2+\beta\) is equal to
[JEE Main 2024, 8 Apr (Shift 1)]
171
\(H:\frac{{y}^{2}}{{b}^{2}}-\frac{{x}^{2}}{{a}^{2}}=1,e=\sqrt{3}\)
\(e=\sqrt{1+\frac{{a}^{2}}{{b}^{2}}}=\sqrt{3}\Rightarrow \frac{{a}^{2}}{{b}^{2}}=2\)
\({a}^{2}=2{b}^{2}\)
length of L.R. \(=\frac{2 a^2}{b}=4 \sqrt{3}\)
\(P(\alpha, 6)\) lie on \(\frac{y^2}{3}-\frac{x^2}{6}=1\)
\(12-\frac{{\alpha }^{2}}{6}=1\Rightarrow {\alpha }^{2}=66\)
Foci \(=(0, \pm b e)=(0,3), (0,-3)\)
Let \({d}_{1},{d}_{2}\) be focal distances of \(P(\alpha ,6)\)
\({d}_{1}=\sqrt{{\alpha }^{2}+(6+be{)}^{2}},{d}_{2}=\sqrt{{\alpha }^{2}+(6-be{)}^{2}}\)
\({d}_{1}=\sqrt{66+81},{d}_{2}=\sqrt{66+9}\)
\(\beta ={d}_{1}{d}_{2}=\sqrt{147\times 75}=105\)
\({\alpha }^{2}+\beta =66+105=171\)
Let \(H: \frac{-x^2}{a^2}+\frac{y^2}{b^2}=1\) be the hyperbola, whose eccentricity is \(\sqrt{3}\) and the length of the latus rectum is \(4 \sqrt{3}\). Suppose the point \((\alpha, 6), \alpha>0\) lies on \(H\). If \(\beta\) is the product of the focal distances of the point \((\alpha, 6)\), then \(\alpha^2+\beta\) is equal to
[JEE Main 2024, 8 Apr (Shift 1)]
171
\(H:\frac{{y}^{2}}{{b}^{2}}-\frac{{x}^{2}}{{a}^{2}}=1,e=\sqrt{3}\)
\(e=\sqrt{1+\frac{{a}^{2}}{{b}^{2}}}=\sqrt{3}\Rightarrow \frac{{a}^{2}}{{b}^{2}}=2\)
\({a}^{2}=2{b}^{2}\)
length of L.R. \(=\frac{2 a^2}{b}=4 \sqrt{3}\)
\(P(\alpha, 6)\) lie on \(\frac{y^2}{3}-\frac{x^2}{6}=1\)
\(12-\frac{{\alpha }^{2}}{6}=1\Rightarrow {\alpha }^{2}=66\)
Foci \(=(0, \pm b e)=(0,3), (0,-3)\)
Let \({d}_{1},{d}_{2}\) be focal distances of \(P(\alpha ,6)\)
\({d}_{1}=\sqrt{{\alpha }^{2}+(6+be{)}^{2}},{d}_{2}=\sqrt{{\alpha }^{2}+(6-be{)}^{2}}\)
\({d}_{1}=\sqrt{66+81},{d}_{2}=\sqrt{66+9}\)
\(\beta ={d}_{1}{d}_{2}=\sqrt{147\times 75}=105\)
\({\alpha }^{2}+\beta =66+105=171\)
Let for two distinct values of \(p\) the lines \(y = x + p\) touch the ellipse E : \(\frac{{\mathrm{x}}^{2}}{{4}^{2}}+\frac{{\mathrm{y}}^{2}}{{3}^{2}}=1\) at the points \(A\) and \(B\). Let the line \(y = x\) intersect \(E\) at the points \(C\) and \(D\) . Then the area of the quadrilateral \(ABCD\) is equal to
[JEE Main 2025, 4 Apr (Shift 2)]
\(24\)
Point of contact are \(\left(\frac{∓{a}^{2}m}{\sqrt{{a}^{2}{m}^{2}+{b}^{2}}},\frac{\pm {b}^{2}}{\sqrt{{a}^{2}{m}^{2}+{b}^{2}}}\right)\)
\(\mathrm{A}\left(\frac{-16}{5},\frac{9}{5}\right)\mathrm{and}\mathrm{B}\left(\frac{16}{5},\frac{-9}{5}\right)\)
Point \(D\) is \(\left(\frac{12}{5},\frac{12}{5}\right)\)
Area of \(ABD=\frac{1}{2}\left|\begin{matrix}-\frac{16}{5} & \frac{9}{5} & 1 \\ \frac{16}{5} & \frac{-9}{5} & 1 \\ \frac{12}{5} & \frac{12}{5} & 1\end{matrix}\right|=12\)
Area of \(ABCD\) is \(=24\)
If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse is :
[JEE Main 2025, 2 Apr (Shift 2)]
\(\frac{4}{\sqrt{17}}\)
Length of minor axis \(=2 b=\frac{1}{4}(2 a e)\)
\(\Rightarrow 4b=ae\Rightarrow 16{b}^{2}={a}^{2}{e}^{2}\)
\(\Rightarrow 16\left({a}^{2}-{a}^{2}{e}^{2}\right)={a}^{2}{e}^{2}\Rightarrow 16{a}^{2}=17{a}^{2}{e}^{2}\)
\(\Rightarrow {e}^{2}=\frac{16}{17}\Rightarrow e=\frac{4}{\sqrt{17}}\)
Let \(f(x)=x^2+9, g(x)=\frac{x}{x-9}\) and \(a =f \circ g(10), b =g \circ f(3)\). If \(e\) and \(l\) denote the eccentricity and the length of the latus rectum of the ellipse \(\frac{x^2}{a}+\frac{y^2}{b}=1\), then \(8 e ^2+l^2\) is equal to.
[JEE Main 2024, 09 Apr (Shift 1)]
8
\(f\left(x\right)={x}^{2}+9,g\left(x\right)=\frac{x}{x-9}\)
\(a=f\left(g\left(10\right)\right)=f\left(\frac{10}{10-9}\right)\)
\(=f\left(10\right)=109\)
\(b=g\left(f\left(3\right)\right)=g\left(9+9\right)\)
\(=g\left(18\right)=\frac{18}{9}=2\)
\(E:\frac{{x}^{2}}{109}+\frac{{y}^{2}}{2}=1\)
\({\mathrm{e}}^{2}=1-\frac{2}{109}=\frac{107}{109}\)
\(ℓ=\frac{2(2)}{\sqrt{109}}=\frac{4}{\sqrt{109}}\)
\(8{\mathrm{e}}^{2}+{ℓ}^{2}=\frac{8(107)}{109}+\frac{16}{109}\)
\(=8\)
Let \(f(x)=x^2+9, g(x)=\frac{x}{x-9}\) and \(a =f \circ g(10), b =g \circ f(3)\). If \(e\) and \(l\) denote the eccentricity and the length of the latus rectum of the ellipse \(\frac{x^2}{a}+\frac{y^2}{b}=1\), then \(8 e ^2+l^2\) is equal to.
[JEE Main 2024, 09 Apr (Shift 1)]
8
\(f\left(x\right)={x}^{2}+9,g\left(x\right)=\frac{x}{x-9}\)
\(a=f\left(g\left(10\right)\right)=f\left(\frac{10}{10-9}\right)\)
\(=f\left(10\right)=109\)
\(b=g\left(f\left(3\right)\right)=g\left(9+9\right)\)
\(=g\left(18\right)=\frac{18}{9}=2\)
\(E:\frac{{x}^{2}}{109}+\frac{{y}^{2}}{2}=1\)
\({\mathrm{e}}^{2}=1-\frac{2}{109}=\frac{107}{109}\)
\(ℓ=\frac{2(2)}{\sqrt{109}}=\frac{4}{\sqrt{109}}\)
\(8{\mathrm{e}}^{2}+{ℓ}^{2}=\frac{8(107)}{109}+\frac{16}{109}\)
\(=8\)
Let the point \(P\) of the focal chord \(PQ\) of the parabola \({y}^{2}=16x\) be \((1,-4)\). If the focus of the parabola divides the chord \(PQ\) in the ratio \(m:n\), \(\gcd (\mathrm{m},\mathrm{n})=1\), then \({m}^{2}+{n}^{2}\) is equal to :
[JEE Main 2025, 2 Apr (Shift 2)]
\(17\)
\(P(a{t}^{2},2at)\)
\(\Rightarrow P(4{t}^{2},8t)\)
\(=(1,-4)\)
\(\Rightarrow t=\frac{−1}{2}\);\(Q\left(\frac{a}{{t}^{2}},\frac{−2a}{t}\right)\)
\(S(4,0)\) is the focus & \(PS=a+{a}^{2}\)
\(QS=a+\frac{a}{{t}^{2}}\)
\(\frac{PS}{QS}={t}^{2}=\frac{4}{1}=\frac{{m}^{}}{n}\)
\({m}^{2}+{n}^{2}=17\)
Let \(P\) be the point on the parabola \(y = x^2\) such that the slope of the tangent to the parabola at the point \(P\) is \(4\) . Let Q be the point in the first quadrant lying on the circle \({x}^{2}+{y}^{2}=2\) such that the slope of the tangent to the circle at the point \(Q\) is \(-1\). Let \(R\) be the point in the first quadrant lying on the ellipse \({x}^{2}+4{y}^{2}=8\) such that the slope of the tangent to the ellipse at the point \(R\) is \(−\frac{1}{2}\). Then the radius of the circle passing through the points \(P, Q\) and \(R\) is
[JEE Advanced 2026]
\(\sqrt{\frac{5}{2}}\)
For \(y=x^2\), slope \(=2x\).
\(2x=4\Rightarrow x=2\), so \(P=(2,4)\).
For \(x^2+y^2=2\), slope \(=-\dfrac{x}{y}\).
\(-\dfrac{x}{y}=-1\Rightarrow x=y\).
Since \(x^2+y^2=2\), we get \(2x^2=2\Rightarrow x=1\), so \(Q=(1,1)\).
For \(x^2+4y^2=8\), slope \(=-\dfrac{x}{4y}\).
\(-\dfrac{x}{4y}=-\dfrac12\Rightarrow x=2y\).
\((2y)^2+4y^2=8\Rightarrow 8y^2=8\Rightarrow y=1\), so \(R=(2,1)\).
Let centre of circle be \(C(h,k)\).
Since \(P=(2,4)\) and \(R=(2,1)\), perpendicular bisector of \(PR\) is \(y=\dfrac52\).
So centre is \(C(h,\dfrac52)\).
Using \(CQ=CR\):
\((h-1)^2+\left(\dfrac52-1\right)^2=(h-2)^2+\left(\dfrac52-1\right)^2\)
\((h-1)^2=(h-2)^2\Rightarrow h=\dfrac32\)
Radius \(=CR=\sqrt{\left(\dfrac32-2\right)^2+\left(\dfrac52-1\right)^2}\)
\(r=\sqrt{\dfrac14+\dfrac94}=\sqrt{\dfrac{10}{4}}=\sqrt{\dfrac52}\)
Let for two distinct values of p the lines y = x + p touch the ellipse E : \(\frac{{\mathrm{x}}^{2}}{{4}^{2}}+\frac{{\mathrm{y}}^{2}}{{3}^{2}}=1\) at the points A and B. Let the line y = x intersect E at the points C and D . Then the area of the quadrilateral ABCD is equal to
[JEE Main 2025, 4 Apr (Shift 1)]
\(24\)
\(\left(\frac{∓\text{ }{a}^{2}m}{\sqrt{{a}^{2}\text{ }{m}^{2}+{b}^{2}}},\text{ }\frac{\pm {b}^{2}}{\sqrt{{a}^{2}\text{ }{m}^{2}+{b}^{2}}}\right)\)
\(A\left(\frac{−16}{5},\text{ }\frac{9}{5}\right)\text{ }B\left(\frac{16}{5},\text{ }\frac{−9}{5}\right)\)
Point D is \(\left(\frac{12}{5},\text{ }\frac{12}{5}\right)\)
Area of \(ABD=\text{ }\frac{1}{2}\text{ }\left|\begin{matrix}-\frac{16}{5} & \frac{9}{5} & 1 \\ \frac{16}{5} & \frac{-9}{5} & 1 \\ \frac{12}{5} & \frac{12}{5} & 1\end{matrix}\right|\)
= 12
Area of ABCD is 24
Let \(\mathrm{E}: \frac{\mathrm{x}^2}{\mathrm{a}^2}+\frac{\mathrm{y}^2}{\mathrm{~b}^2}=1, \mathrm{a}>\mathrm{b}\) and \(\mathrm{H}: \frac{\mathrm{x}^2}{\mathrm{~A}^2}-\frac{\mathrm{y}^2}{\mathrm{~B}^2}=1\).
Let the distance between the foci of E and the foci of H be \(2 \sqrt{3}\). If \(\mathrm{a}-\mathrm{A}=2\), and the ratio of the eccentricities of E and H is \(\frac{1}{3}\), then the sum of the lengths of their latus rectums is equal to :
[JEE Main 2025, 22 Jan (Shift 2)]
8
\(\text{ }\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\text{ foci are }(\pm a{e}_{1},0)\\ \frac{{x}^{2}}{{A}^{2}}-\frac{{y}^{2}}{{B}^{2}}=1\text{ foci are }\left(\pm A{e}_{2},0\right)\\ \Rightarrow 2{\mathrm{ae}}_{1}=2\sqrt{3}\Rightarrow {\mathrm{ae}}_{1}=\sqrt{3}\\ \text{ and }2A{e}_{2}=2\sqrt{3}\Rightarrow A{e}_{2}=\sqrt{3}\\ \text{and}2{\mathrm{ae}}_{1}=2A{e}_{2}\Rightarrow \frac{{e}_{1}}{{e}_{2}}=\frac{A}{a}=\frac{1}{3}\left(\text{Given}\right)\\ \Rightarrow a=3A..\text{.}\left(1\right)\\ \text{ Now }a-A=2\Rightarrow a-\frac{a}{3}=2\\ \Rightarrow a=3\text{ and }A=1\\ a{e}_{1}=\sqrt{3}\Rightarrow {\mathrm{e}}_{1}=\frac{1}{\sqrt{3}}\text{ and }{\mathrm{e}}_{2}=\sqrt{3}\\ {\mathrm{b}}^{2}={\mathrm{a}}^{2}\left(1-{{\mathrm{e}}_{1}}^{2}\right)\\ {\mathrm{b}}^{2}=6\\ \text{ and }{\mathrm{B}}^{2}={\mathrm{A}}^{2}\left({\left({\mathrm{e}}_{2}\right)}^{2}-1\right)=(2)\Rightarrow {\mathrm{B}}^{2}=2\\ \text{sum of }\mathrm{LR}=\frac{2{\mathrm{b}}^{2}}{\mathrm{a}}+\frac{2{\mathrm{B}}^{2}}{\mathrm{A}}=8\\\)
If the eccentricity \(e\) of the hyperbola \(\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1\), passing through \((6,4\sqrt{3})\), satisfies \(15\left({e}^{2}+1\right)=34e\), then the length of the latus rectum of the hyperbola \(\frac{{x}^{2}}{{b}^{2}}-\frac{{y}^{2}}{2\left({a}^{2}+1\right)}=1\) is:
[JEE Main 2026, 6 Apr (Shift 1)]
\(10\)
\(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\)
It passes through \((6, 4\sqrt{3})\).
\(\Rightarrow \frac{36}{a^2}-\frac{48}{b^2}=1\)\(\ldots(1)\)
Also, \(15 \mathrm{e}^2-34 \mathrm{e}+15=0\)
\(\Rightarrow 15 \mathrm{e}^2-25 \mathrm{e}-9 \mathrm{e}+15=0\)
\(\mathrm{e}=\frac{5}{3}\) or \(\frac{3}{5} \Rightarrow \mathrm{e}=\frac{5}{3}\)
\(\Rightarrow \mathrm{e}^{2}=\frac{25}{9}\)
\(1+\frac{b^2}{a^2}=\frac{25}{9} \Rightarrow \frac{b^2}{a^2}=\frac{16}{9}\) \(\ldots(2)\)
Using (1) and (2):
\(\Rightarrow \frac{36}{a^2}-\frac{48}{16 a^2} \times 9=1 \Rightarrow a=3, b=4\)
Now, the length of the latus rectum of the hyperbola \(\frac{x^2}{b^2}-\frac{y^2}{2\left(a^2+1\right)}=1\) is:
\(=\frac{4\left(a^2+1\right)}{b}=\frac{4(10)}{4}=10\)
Let the image of parabola \(x^2=4 y\), in the line \(x-y=1\) be \((y+a)^2=b(x-c), a, b, c \in N\). Then \(a+b+c\) is equal to
[JEE Main 2026, 24 Jan (Shift 2)]
\(6\)
Parametric point \(P\) on \(x^2=4 y\) is \(P\left(2 t, t^2\right)\)
∴ mirror image of \(P\) in \(x-y=1\) is
\(Q\equiv \left(2t−\frac{2⋅1⋅\left(2t−{t}^{2}−1\right)}{2},{t}^{2}+\frac{2⋅1⋅\left(2t−{t}^{2}−1\right)}{2}\right)\)
\(Q\equiv \left({t}^{2}+1,2t−1\right)\equiv (h,k)\)
So, \(h=t^2+1, k=2t-1\)
\(\Rightarrow t=\frac{k+1}{2}\)
\(\Rightarrow h= \frac{(k+1)^2}{4} +1\)
∴ locus of \(Q\) is \(x=\frac{{(y+1)}^{2}}{4}+1\) which is the required parabola.
\(∴{(y+1)}^{2}=4\left(x−1\right)\)
\(∴a=1,b=4,c=1\)
\(∴a+b+c=6\)
Let an ellipse \(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1,a [JEE Main 2026, 2 Apr (Shift 1)]
\(\frac{8\sqrt{5}}{3}\)
Given \( \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \)
passes \((4,3) \)
\( \frac{16}{a^2}+\frac{9}{b^2}=1 \ldots \)(1)
\(e=\frac{\sqrt{5}}{3} \)
\( e^2=\frac{5}{9}\)
\( 1-\frac{a^2}{b^2}=\frac{5}{9} \)
\( \frac{a^2}{b^2}=\frac{4}{9} \ldots\)(2)
From (1) and (2)
\( \frac{16}{a^2}+\frac{4}{a^2}=1\)
\( \frac{20}{a^2}=1 \Rightarrow a^2=20 \ \\& \ b^2=45\)
\(\ell(L R)=\frac{2 a^2}{b}=\frac{2 \times 20}{\sqrt{45}}=\frac{40}{\sqrt{45}}\)
\( =\frac{40 \sqrt{5}}{3 \sqrt{5} \times \sqrt{5}}=\frac{8 \sqrt{5}}{3}\)
Let the parabola \(y={x}^{2}+px+q\) passing through the point \((1,-1)\) be such that the distance between its vertex and the \(x-\)axis is minimum. Then the value of \({p}^{2}+{q}^{2}\) is:
[JEE Main 2026, 2 Apr (Shift 2)]
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The equation of the chord, of the ellipse \(\frac{x^2}{25}+\frac{y^2}{16}=1\), whose mid-point is \((3,1)\) is :
[JEE Main 2025, 24 Jan (Shift 2)]
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Let one root of the quadratic equation in \(x\):
\(\left({k}^{2}-15k+27\right){x}^{2}+9(k-1)x+18=0\)
be twice the other. Then the length of the latus rectum of the parabola \({y}^{2}=6kx\) is equal to:
[JEE Main 2026, 6 Apr (Shift 1)]
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Let \(\mathrm{E}: \frac{\mathrm{x}^2}{\mathrm{a}^2}+\frac{\mathrm{y}^2}{\mathrm{~b}^2}=1, \mathrm{a}>\mathrm{b}\) and \(\mathrm{H}: \frac{\mathrm{x}^2}{\mathrm{~A}^2}-\frac{\mathrm{y}^2}{\mathrm{~B}^2}=1\).
Let the distance between the foci of E and the foci of H be \(2 \sqrt{3}\). If \(\mathrm{a}-\mathrm{A}=2\), and the ratio of the eccentricities of E and H is \(\frac{1}{3}\), then the sum of the lengths of their latus rectums is equal to :
[JEE Main 2025, 22 Jan (Shift 2)]
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Let one focus of the hyperbola \(\mathrm{H}:\frac{{\mathrm{x}}^{2}}{{\mathrm{a}}^{2}}-\frac{{\mathrm{y}}^{2}}{{\mathrm{b}}^{2}}=1\) be at \((\sqrt{10},0)\) and the corresponding directrix be \(\mathrm{x}=\frac{9}{\sqrt{10}}\). If \(e\) and \(l\) respectively are the eccentricity and the length of the latus rectum of \(H\) , then \(9\left({\mathrm{e}}^{2}+l\right)\) is equal to:
[JEE Main 2025, 2 Apr (Shift 1)]
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Let the parabola \(y=x^2+p x-3\) cuts the coordinate axes at P , \(\mathbf{Q}\) and \(\mathbf{R}\). A circle with centre \(( -1,-1 )\) passes through \(\mathbf{P}, \mathbf{Q}\) and \(R\), then the area of triangle PQR. (24 Jan, Shift I, Memory Based)
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Let the length of the latus rectum of an ellipse \(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\), \((a > b)\), be \(30\). If its eccentricity is the maximum value of the function \(f\left(t\right)=−\frac{3}{4}+2t−{t}^{2}\), then \(\left(a^2+b^2\right)\) is equal to
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The foci of the hyperbola \(4 x^2-9 y^2-1=0\) are:
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Let the product of the focal distances of the point \(\left(\sqrt{3},\frac{1}{2}\right)\text{ on the ellipse }\frac{{\mathrm{x}}^{2}}{{\mathrm{a}}^{2}}+\frac{{\mathrm{y}}^{2}}{{\mathrm{b}}^{2}}=1,(\mathrm{a}>\mathrm{b})\text{, be }\frac{7}{4}.\) Then the absolute difference of the eccentricities of two such ellipses is
[JEE Main 2025, 24 Jan (Shift 1)]
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If the focus of the parabola
\((y-k)^2=4(x-h)\)
always lies between the lines \(x+y=1\) and \(x+y=3\) then:
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Let the ellipse, \({\mathrm{E}}_{1}:\frac{{\mathrm{x}}^{2}}{{\mathrm{a}}^{2}}+\frac{{\mathrm{y}}^{2}}{{\mathrm{b}}^{2}}=1,\mathrm{a}>\mathrm{b}\) and \({\mathrm{E}}_{2}:\frac{{\mathrm{x}}^{2}}{{\mathrm{A}}^{2}}+\frac{{\mathrm{y}}^{2}}{{\mathrm{B}}^{2}}=1,\mathrm{A}<\mathrm{B}\) have same eccentricity \(\frac{1}{\sqrt{3}}.\) Let the product of their lengths of latus rectums be \(\frac{32}{\sqrt{3}},\) and the distance between the foci of \(E_1\) be \(4.\) If \(E_1\) and \(E_2\) meet at \(A,B,C\) and \(D,\) then the area of the quadrilateral \(ABCD\) equals:
[JEE Main 2025, 29 Jan (Shift 1)]
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Let the foci of a hyperbola be \((1, 14)\) and \((1, –12).\) If it passes through the point \((1, 6),\) then the length of its latus-rectum is :
[JEE Main 2025, 22 Jan (Shift 1)]
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Let the domain of the function \(f(x)={\log }_{3}{\log }_{5}{\log }_{7}\left(9x-{x}^{2}-13\right)\) be the interval \((m, n)\). Let the hyperbola \(\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1\) have eccentricity \(\frac{n}{3}\) and the length of the latus rectum \(\frac{8m}{3}\). Then \({b}^{2}-{a}^{2}\) is equal to:
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If the equation of the parabola with vertex \(\mathrm{V}\left(\frac{3}{2}, 3\right)\) and the directrix \(x+2 y=0\) is\(\alpha x^2+\beta y^2-\gamma x y-30 x-60 y+225=0\), then \(\alpha+\beta+\gamma\)is equal to :
[JEE Main 2025, 24 Jan (Shift 2)]
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\({x}^{2}+{y}^{2}-8x=0,\frac{{x}^{2}}{9}-\frac{{y}^{2}}{4}=1\) intersect at A, B.
A triangle is formed using vertices A, B, C where C lies on
\(2x-3y+4=0\). find locus of centroid of \(∆\mathrm{ABC}\).
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The equation of chord of the ellipse \(\frac{x^2}{25}+\frac{y^2}{16}=1\) with \((3,1)\) as mid-point is (24 Jan, Shift II, Memory Based)
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Let the domain of the function \(f(x)={\log }_{3}{\log }_{5}{\log }_{7}\left(9x-{x}^{2}-13\right)\) be the interval \((m, n)\). Let the hyperbola \(\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1\) have eccentricity \(\frac{n}{3}\) and the length of the latus rectum \(\frac{8m}{3}\). Then \({b}^{2}-{a}^{2}\) is equal to:
[JEE Main 2026, 23 Jan (Shift 2)]
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Let the directrix of the parabola \(\mathrm{P}: \mathrm{y}^2=8 x\), cut \(x\)-axis at the point \(A\) . Let \(\mathrm{B}(\alpha, \beta), \alpha>1\), be a point on \(P\) such that the slope of \(A B\) is \(\frac{3}{5}\). If \(B C\) is a focal chord of \(P\), then six times the area of \(\triangle A B C\) is:
[JEE Main 2026, 5 Apr (Shift 2)]
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If the line \(\alpha x+2y=1\), where \(\alpha \in ℝ\), does not meet the hyperbola \({x}^{2}−9{y}^{2}=9\), then a possible value of \(\alpha\) is:
[JEE Main 2026, 22 Jan (Shift 1)]
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\({x}^{2}+{y}^{2}-8x=0,\frac{{x}^{2}}{9}-\frac{{y}^{2}}{4}=1\) intersect at A, B.
A triangle is formed using vertices A, B, C where C lies on
\(2x-3y+4=0\). find locus of centroid of \(∆\mathrm{ABC}\).
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Let the length of a latus rectum of an ellipse \(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\) be \(10\). If its eccentricity is the minimum value of the function \(f(\mathrm{t})={\mathrm{t}}^{2}+\mathrm{t}+\frac{11}{12}\), \(\mathrm{t}\in R\), then \({\mathrm{a}}^{2}+{\mathrm{b}}^{2}\) is equal to :
[JEE Main 2025, 7 Apr (Shift 2)]
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Let \({e}_{1}\) and \({e}_{2}\) be the eccentricities of the ellipse \(\frac{{x}^{2}}{{b}^{2}}+\frac{{y}^{2}}{25}=1\) and the hyperbola \(\frac{{x}^{2}}{16}-\frac{{y}^{2}}{{b}^{2}}=1\), respectively. If \(\mathrm{b}<5\) and \({\mathrm{e}}_{1}{\mathrm{e}}_{2}=1\), then the eccentricity of the ellipse having its axes along the coordinate axes and passing through all four foci (two of the ellipse and two of the hyperbola) is :
[JEE Main 2025, 7 Apr (Shift 2)]
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Distance between the focus of standard ellipse and hyperbola is \(2 \sqrt{3}\), ratios of there eccentricity being 1:3. Difference of length of their semi-major axis and semi transverse axis is 2 then find sum of length of their latus rectum? (22 Jan, Shift II, Memory Based)
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Let each of the two ellipses \({\text{E}}_{1}:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1,\left(a>b\right)\) and \({\text{E}}_{2}:\frac{{x}^{2}}{{\text{ A}}^{2}}+\frac{{y}^{2}}{{\text{ B}}^{2}}=1,\left(\text{A}<\text{B}\right)\) have eccentricity \(\frac{4}{5}.\) Let the lengths of the latus rectum of \(E_1\) and \(E_2\) be \(l_1\) and \(l_2\) , respectively, such that \(2{l}_{1}^{2}=9{l}_{2}\). If the distance between the foci of \(E_1\) is 8, then the distance between the foci of \(E_2\) is
[JEE Main 2026, 24 Jan (Shift 1)]
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Distance between the focus of standard ellipse and hyperbola is \(2 \sqrt{3}\), ratios of there eccentricity being 1:3. Difference of length of their semi-major axis and semi transverse axis is 2 then find sum of length of their latus rectum? (22 Jan, Shift II, Memory Based)
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Find length of chord whose midpoint is \(\ (1/2, 1) \) of ellipse \(\frac{x^2}{2}+\frac{y^2}{4}=1\)
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If the equation of the parabola with vertex \(\mathrm{V}\left(\frac{3}{2}, 3\right)\) and the directrix \(x+2 y=0\) is\(\alpha x^2+\beta y^2-\gamma x y-30 x-60 y+225=0\), then \(\alpha+\beta+\gamma\) is equal to :
[JEE Main 2025, 24 Jan (Shift 2)]
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If the focus of the parabola
\((y-k)^2=4(x-h)\)
always lies between the lines \(x+y=1\) and \(x+y=3\) then:
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The foci of the hyperbola \(4 x^2-9 y^2-1=0\) are:
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Let one focus of the hyperbola \(\mathrm{H}:\frac{{\mathrm{x}}^{2}}{{\mathrm{a}}^{2}}-\frac{{\mathrm{y}}^{2}}{{\mathrm{b}}^{2}}=1\) be at \((\sqrt{10},0)\) and the corresponding directrix be \(\mathrm{x}=\frac{9}{\sqrt{10}}\). If e and \(l\) respectively are the eccentricity and the length of the latus rectum of H , then \(9\left({\mathrm{e}}^{2}+l\right)\) is equal to:
[JEE Main 2025, 2 Apr (Shift 1)]
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If \(A\) and \(B\) are the points of intersection of the circle \(x^2+y^2-8 x=0\) and the hyperbola \(\frac{x^2}{9}-\frac{y^2}{4}=1\) and a point \(P\) moves on the line \(2 x-3 y+4=0\), then the centroid of \(\triangle P A B\) lies on the line :
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A rod of length 8 units having two end points always lie on \(x-y+2=0\) and \(x+y+2=0\). A point \(P\) divide this line in ratio \(2: 1\). Then locus of \(P\) is
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If the line \(\alpha x+2y=1\), where \(\alpha \in ℝ\), does not meet the hyperbola \({x}^{2}−9{y}^{2}=9\), then a possible value of \(\alpha\) is:
[JEE Main 2026, 22 Jan (Shift 1)]
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Let \(A\) be the focus of the parabola \(y^2=8 x\). Let the line \(y=m x+c\) intersect the parabola at two distinct points \(B\) and \(C\) . If the centroid of the \(\triangle A B C\) is \(\left(\frac{7}{3}, \frac{4}{3}\right)\), then \((B C)^2\) is equal to:
[JEE Main 2026, 28 Jan (Shift 2)]
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The length of the chord of the ellipse \(\frac{x^2}{4}+\frac{y^2}{2}=1\), whose mid-point is \(\left(1, \frac{1}{2}\right)\), is:
[JEE Main 2025, 23 Jan (Shift 2)]
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Consider a hyperbola H having centre at the origin and foci on the x-axis. Let \({C}_{1}\) be the circle touching the hyperbola H and having the centre at the origin. Let \({C}_{2}\) be the circle touching the hyperbola H at its vertex and having the centre at one of its foci. If areas (in sq units) of \({C}_{1}\)and \({C}_{2}\) are \(36\pi\) and \(4\pi\), respectively, then the length (in units) of latus rectum of H isEndFragment
[JEE Main 2024, 04 Apr (Shift 2)]
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Consider a hyperbola H having centre at the origin and foci on the x-axis. Let \({C}_{1}\) be the circle touching the hyperbola H and having the centre at the origin. Let \({C}_{2}\) be the circle touching the hyperbola H at its vertex and having the centre at one of its foci. If areas (in sq units) of \({C}_{1}\)and \({C}_{2}\) are \(36\pi\) and \(4\pi\), respectively, then the length (in units) of latus rectum of H isEndFragment
[JEE Main 2024, 04 Apr (Shift 2)]
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Let \(O\) be the origin, and \(P\) and \(Q\) be two points on the rectangular hyperbola \(xy=12\) such that the midpoint of the line segment \(PQ\) is \(\left(\frac{1}{2},-\frac{1}{2}\right)\). Then the area of the triangle \(OPQ\) equals :
[JEE Main 2026, 2 Apr (Shift 2)]
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Let \(P(10,2 \sqrt{15})\) be a point on the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\), whose foci are \(S\) and \(S^{\prime}\). If the length of its latus rectum is \(8\), then the square of the area of \(\triangle P S S^{\prime}\) is equal to
[JEE Main 2026, 22 Jan (Shift 2)]
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Let the sum of the focal distances of the point \(\mathrm{P}(4,3)\) on the hyperbola \(\mathrm{H}:\frac{{\mathrm{x}}^{2}}{{\mathrm{a}}^{2}}-\frac{{\mathrm{y}}^{2}}{{\mathrm{b}}^{2}}=1\) be \(8\sqrt{\frac{5}{3}}\). If for H , the length of the latus rectum is \(l\) and the product of the focal distances of the point \(P\) is \(m\) , then \(9{l}^{2}+6\mathrm{m}\) is equal to :-
[JEE Main 2025, 4 Apr (Shift 2)]
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Let the image of parabola \(x^2=4 y\), in the line \(x-y=1\) be \((y+a)^2=b(x-c), a, b, c \in N\). Then \(a+b+c\) is equal to
[JEE Main 2026, 24 Jan (Shift 2)]
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Let the ellipse \(3{\mathrm{x}}^{2}+{\mathrm{py}}^{2}=4\) pass through the centre \(C\)of the circle \({\mathrm{x}}^{2}+{\mathrm{y}}^{2}-2\mathrm{x}-4\mathrm{y}-11=0\) of radius \(r\). Let \({f}_{1},{f}_{2}\) be the focal distances of the point \(C\)on the ellipse. Then \(6{\mathrm{f}}_{1}{\mathrm{f}}_{2}-\mathrm{r}\) is equal to
[JEE Main 2025, 8 Apr (Shift 1)]
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Let \(S\) and \({S}^{'}\) be the foci of the ellipse \(\frac{{x}^{2}}{25}+\frac{{y}^{2}}{9}=1\) and \(P(\alpha ,\text{ }\beta )\) be a point on the ellipse in the first quadrant. If \({(SP)}^{2}+{({S}^{'}P)}^{2}−SP⋅{S}^{'}P=37,\) then \({\alpha }^{2}+{\beta }^{2}\) is equal to
[JEE Main 2026, 22 Jan (Shift 2)]
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Let \(x=9\) be a directrix of an ellipse \(E\), whose centre is at the origin and eccentricity is \(\frac{1}{3}\). Let \(P(\alpha ,0),\alpha >0\), be a focus of \(E\) and \(A B\) be a chord passing through \(P\). Then the locus of the mid-point of \(AB\) is:
[JEE Main 2026, 6 Apr (Shift 2)]
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Let \({e}_{1}\) and \({e}_{2}\) be the eccentricities of the ellipse \(\frac{{x}^{2}}{{b}^{2}}+\frac{{y}^{2}}{25}=1\) and the hyperbola \(\frac{{x}^{2}}{16}-\frac{{y}^{2}}{{b}^{2}}=1\), respectively. If \(\mathrm{b}<5\) and \({\mathrm{e}}_{1}{\mathrm{e}}_{2}=1\), then the eccentricity of the ellipse having its axes along the coordinate axes and passing through all four foci (two of the ellipse and two of the hyperbola) is :
[JEE Main 2025, 7 Apr (Shift 2)]
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Let the length of the latus rectum of an ellipse \(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\), \((a > b)\), be \(30\). If its eccentricity is the maximum value of the function \(f\left(t\right)=−\frac{3}{4}+2t−{t}^{2}\), then \(\left(a^2+b^2\right)\) is equal to
[JEE Main 2026, 24 Jan (Shift 2)]
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Let the parabola \(y=x^2+p x-3\) cuts the coordinate axes at P , \(\mathbf{Q}\) and \(\mathbf{R}\). A circle with centre \(( -1,-1 )\) passes through \(\mathbf{P}, \mathbf{Q}\) and \(R\), then the area of triangle PQR. (24 Jan, Shift I, Memory Based)
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If \(A\) and \(B\) are the points of intersection of the circle \(x^2+y^2-8 x=0\) and the hyperbola \(\frac{x^2}{9}-\frac{y^2}{4}=1\) and a point \(P\) moves on the line \(2 x-3 y+4=0\), then the centroid of \(\triangle P A B\) lies on the line :
[JEE Main 2025, 28 Jan (Shift 2)]
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Let \(P\) be a point on the ellipse \(\frac{x^2}{9}+\frac{y^2}{4}=1\). Let the line passing through \(P\) and parallel to \(y\)-axis meet the circle \(x^2+y^2=9\) at point \(Q\) such that \(P\) and \(Q\) are on the same side of the \(x\)-axis. Then, the eccentricity of the locus of the point \(R\) on \(P Q\) such that \(P R: R Q=4: 3\) as \(P\) moves on the ellipse, is :EndFragment
[JEE Main 2024, 01 Feb (Shift 2)]
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Let \(P\) be a point on the ellipse \(\frac{x^2}{9}+\frac{y^2}{4}=1\). Let the line passing through \(P\) and parallel to \(y\)-axis meet the circle \(x^2+y^2=9\) at point \(Q\) such that \(P\) and \(Q\) are on the same side of the \(x\)-axis. Then, the eccentricity of the locus of the point \(R\) on \(P Q\) such that \(P R: R Q=4: 3\) as \(P\) moves on the ellipse, is :EndFragment
[JEE Main 2024, 01 Feb (Shift 2)]
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Let the eccentricity \(e\) of a hyperbola satisfy the equation \(6 \mathrm{e}^2-11 \mathrm{e} +3=0\). If the foci of the hyperbola are \((3,5)\) and \((3,-4)\), then the length of its latus rectum is:
[JEE Main 2026, 5 Apr (Shift 2)]
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Let \(\mathrm{P}(3\cos \alpha ,2\sin \alpha ),\alpha \neq 0\), be a point on the ellipse \(\frac{{x}^{2}}{9}+\frac{{y}^{2}}{4}=1\), \(Q\) be a point on the circle \({x}^{2}+{y}^{2}-14x-14y+82=0\) and \(R\) be a point on the line \(x+y=5\) such that the centroid of the triangle \(PQR\) is \(\left(2+\cos \alpha ,3+\frac{2}{3}\sin \alpha \right)\). Then the sum of the ordinates of all possible points R is:
[JEE Main 2026, 4 Apr (Shift 2)]
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The equation of latus rectum of a parabola is \(x+y=8\) and the equation of the tangent at the vertex is \(x+y=12\), then length of the latus rectum is
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Let the sum of the focal distances of the point \(\mathrm{P}(4,3)\) on the hyperbola \(\mathrm{H}:\frac{{\mathrm{x}}^{2}}{{\mathrm{a}}^{2}}-\frac{{\mathrm{y}}^{2}}{{\mathrm{b}}^{2}}=1\) be \(8\sqrt{\frac{5}{3}}\). If for H , the length of the latus rectum is \(l\) and the product of the focal distances of the point P is m , then \(9{l}^{2}+6\mathrm{m}\) is equal to :-
[JEE Main 2025, 4 Apr (Shift 2)]
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Let the length of a latus rectum of an ellipse \(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\) be \(10\). If its eccentricity is the minimum value of the function \(f(\mathrm{t})={\mathrm{t}}^{2}+\mathrm{t}+\frac{11}{12}\), \(\mathrm{t}\in \mathrm{R}\), then \({a}^{2}+{b}^{2}\) is equal to:
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A circle is drawn with the two foci of an ellipse \(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\) as the ends of the diameter. What is the equation of the circle?
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Let an ellipse \(\frac{{x}^{2}}{9}+\frac{{y}^{2}}{4}=1\). If Mid point of Chord is \(\left(\sqrt{2},\frac{4}{3}\right)\). If length of this Chord is \(\frac{2\sqrt{\alpha }}{3}\) .Find value of \(\alpha\).
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Let the parabola \(y={x}^{2}+px-3,\) meet the coordinate axes at the points \(P\), \(Q\) and \(R\). If the circle C with centre at \( (-1,-1) \) passes through the points \(P\), \(Q\) and \(R\), then the area of \(\Delta \mathrm{PQR}\) is:
[JEE Main 2025, 22 Jan (Shift 1)]
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Let \(S\) denote the locus of the point of intersection of the pair of lines
\(4 x-3 y=12 \alpha \)
\(4 \alpha x+3 \alpha y=12\)
where \(\alpha\) varies over the set of non-zero real numbers. Let \(T\) be the tangent to \(S\) passing through the points \((p, 0)\) and \((0, q), q>0\), and parallel to the line \(4 x-\frac{3}{\sqrt{2}} y=0\). Then the value of \(p q\) is
[JEE Advanced 2025]
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Let the ellipse \(E:\frac{{x}^{2}}{144}+\frac{{y}^{2}}{169}=1\) and the hyperbola \(H:\frac{{x}^{2}}{16}−\frac{{y}^{2}}{{\lambda }^{2}}=−1\) have the same foci. If \(e\) and \(L\) respectively denote the eccentricity and the length of the latus rectum of \(H\), then the value of \(24(e+L)\)is:
[JEE Main 2026, 28 Jan (Shift 2)]
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Let the ellipse \(E:\frac{{x}^{2}}{144}+\frac{{y}^{2}}{169}=1\) and the hyperbola \(H:\frac{{x}^{2}}{16}−\frac{{y}^{2}}{{\lambda }^{2}}=−1\) have the same foci. If \(e\) and \(L\) respectively denote the eccentricity and the length of the latus rectum of \(H\), then the value of \(24(e+L)\)is:
[JEE Main 2026, 28 Jan (Shift 2)]
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Find length of chord whose midpoint is \(\ (1/2, 1) \) of ellipse \(\frac{x^2}{2}+\frac{y^2}{4}=1\)
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Let \(T\) be the tangent to the parabola \({y}^{2}=16x\) at the point \((64, 32)\). Let \(L\) be the tangent to the same parabola at another point (\({x}_{1},{y}_{1}\)) on the parabola. If \(L\) and \(T\) are perpendicular to each other, then the distance between the point \(\left({x}_{1},{y}_{1}\right)\) and the focus of the parabola, is
[JEE Advanced 2026]
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Let \(S\) and \({S}^{'}\) be the foci of the ellipse \(\frac{{x}^{2}}{25}+\frac{{y}^{2}}{9}=1\) and \(P(\alpha ,\text{ }\beta )\) be a point on the ellipse in the first quadrant. If \({(SP)}^{2}+{({S}^{'}P)}^{2}−SP⋅{S}^{'}P=37,\) then \({\alpha }^{2}+{\beta }^{2}\) is equal to
[JEE Main 2026, 22 Jan (Shift 2)]
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Let \(\mathrm{E}: \frac{\mathrm{x}^2}{\mathrm{a}^2}+\frac{\mathrm{y}^2}{\mathrm{~b}^2}=1, \mathrm{a}>\mathrm{b}\) and \(\mathrm{H}: \frac{\mathrm{x}^2}{\mathrm{~A}^2}-\frac{\mathrm{y}^2}{\mathrm{~B}^2}=1\).
Let the distance between the foci of E and the foci of H be \(2 \sqrt{3}\). If \(\mathrm{a}-\mathrm{A}=2\), and the ratio of the eccentricities of E and H is \(\frac{1}{3}\), then the sum of the lengths of their latus rectums is equal to :
[JEE Main 2025, 22 Jan (Shift 2)]
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If the chord joining the points \(P_1\left(x_1, y_1\right)\) and \(P_2\left(x_2, y_2\right)\) on the parabola \(y^2=12 x\) subtends a right angle at the vertex of the parabola, then \(x_1 x_2-y_1 y_2\) is equal to
[JEE Main 2026, 22 Jan (Shift 1)]
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Let \(\mathrm{E}: \frac{\mathrm{x}^2}{\mathrm{a}^2}+\frac{\mathrm{y}^2}{\mathrm{~b}^2}=1, \mathrm{a}>\mathrm{b}\) and \(\mathrm{H}: \frac{\mathrm{x}^2}{\mathrm{~A}^2}-\frac{\mathrm{y}^2}{\mathrm{~B}^2}=1\).
Let the distance between the foci of \(E\) and the foci of \(H\) be \(2 \sqrt{3}\). If \(\mathrm{a}-\mathrm{A}=2\), and the ratio of the eccentricities of \(E\) and \(H\) is \(\frac{1}{3}\), then the sum of the lengths of their latus rectums is equal to :
[JEE Main 2025, 22 Jan (Shift 2)]
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If the chord joining the points \(P_1\left(x_1, y_1\right)\) and \(P_2\left(x_2, y_2\right)\) on the parabola \(y^2=12 x\) subtends a right angle at the vertex of the parabola, then \(x_1 x_2-y_1 y_2\) is equal to
[JEE Main 2026, 22 Jan (Shift 1)]
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If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse is :
[JEE Main 2025, 2 Apr (Shift 2)]
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The equation of the chord, of the ellipse \(\frac{x^2}{25}+\frac{y^2}{16}=1\), whose mid-point is \((3,1)\) is :
[JEE Main 2025, 24 Jan (Shift 2)]
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Let \(P\)be the parabola, whose focus is \((-2,1)\) and directrix is \(2x+y+2=0\). Then the sum of the ordinates of the points on \(P\), whose abscissa is \(-2\) , is
[JEE Main 2025, 7 Apr (Shift 1)]
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The equation of chord of the ellipse \(\frac{x^2}{25}+\frac{y^2}{16}=1\) with \((3,1)\) as mid-point is (24 Jan, Shift II, Memory Based)
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Let an ellipse \(\frac{{x}^{2}}{9}+\frac{{y}^{2}}{4}=1\). If Mid point of Chord is \(\left(\sqrt{2},\frac{4}{3}\right)\). If length of this Chord is \(\frac{2\sqrt{\alpha }}{3}\) .Find value of \(\alpha\).
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Let the parabola \(y={x}^{2}+px-3,\) meet the coordinate axes at the points P, Q and R. If the circle C with centre at \( (-1,-1) \) passes through the points P, Q and R, then the area of \(\Delta \mathrm{PQR}\) is:
[JEE Main 2025, 22 Jan (Shift 1)]
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If the line \(3x-2y+12=0\) intersects the parabola \(4y=3{x}^{2}\) at the points A and B, then at the vertex of the parabola, the line segment \(AB\) subtends an angle equal to
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Let \(A\) be the focus of the parabola \(y^2=8 x\). Let the line \(y=m x+c\) intersect the parabola at two distinct points \(B\) and \(C\) . If the centroid of the \(\triangle A B C\) is \(\left(\frac{7}{3}, \frac{4}{3}\right)\), then \((B C)^2\) is equal to:
[JEE Main 2026, 28 Jan (Shift 2)]
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If the midpoint of a chord of the ellipse \(\frac{x^2}{9}+\frac{y^2}{4}=1\) is \((\sqrt{2}, \frac{4}{3})\), and the length of the chord is \(\frac{2 \sqrt{\alpha}}{3}\), then \(\alpha\) is :
[JEE Main 2025, 28 Jan (Shift 2)]
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Let \(P(10,2 \sqrt{15})\) be a point on the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\), whose foci are \(S\) and \(S^{\prime}\). If the length of its latus rectum is \(8\), then the square of the area of \(\triangle P S S^{\prime}\) is equal to
[JEE Main 2026, 22 Jan (Shift 2)]
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Let \(e_1\) and \(e_2\) be two distinct roots of the equation \(x^2-a x+2=0\). Let the sets \(\left\{a \in \mathbb{R}: e_1\right.\) and \(e_2\) are the eccentricities of hyperbolas\(\}=(\alpha, \beta)\), and \(\left\{a\in \mathbb{R}:e_1\right.\) and \(e_2\) are the eccentricities of an ellipse and a hyperbola, respectively\(\}=(\gamma, \infty)\). Then \(\alpha^2+\beta^2+\gamma^2\) is equal to:
[JEE Main 2026, 6 Apr (Shift 1)]
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The eccentricity of an ellipse \(E\) with centre at the origin \(O\) is \(\frac{\sqrt{3}}{2}\) and its directrices are \(x= \pm \frac{4 \sqrt{6}}{3}\). Let \(H: \frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) be a hyperbola whose eccentricity is equal to the length of semi-major axis of \(E\) , and whose length of latus rectum is equal to the length of minor axis of \(E\). Then the distance between the foci of \(H\) is:
[JEE Main 2026, 6 Apr (Shift 2)]
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The length of the latus rectum of the parabola which has focus at \((-1,1)\) and the directrix is \(4 x+3 y-24=0\) is
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Let \(A(2\sec { \theta } ,3\tan { \theta } )\) and \(B(2\sec { \phi } ,3\tan { \phi } )\) where \(\theta +\phi =\cfrac { \pi }{ 2 } \), be two points on the hyperbola \(\cfrac { { x }^{ 2 } }{ 4 } -\cfrac { { y }^{ 2 } }{ 9 } =1\). If \(\left( \alpha ,\beta \right) \) is the point of intersection of normals to the hyperbola at \(A\) and \(B\), then \(\beta=\)
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Let \(H\) be the hyperbola, whose foci are \((1 \pm \sqrt{2}, 0)\) and eccentricity is \(\sqrt{2}\). Then the length of its latus rectum is
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Let \(\theta\) be the acute angle between the tangents to the ellipse \(\frac{{x}^{2}}{9}+\frac{{y}^{2}}{1}=1\) and the circle \({x}^{2}+{y}^{2}=3\) at their point of intersection in the first quadrant. The \(\tan \theta\) is equal to:
[JEE Main 2021, 1 Sep (Shift 2)]
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If the three normals drawn to the parabola, \({y}^{2}=2x\) pass through the point \((a, 0), a \neq 0\) then \('a'\) must be greater than:
[JEE Main 2021, 16 Mar (Shift 1)]
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Let the tangent to the parabola \(S:{y}^{2}=2x\) at the point \(P(2,2)\) meet the x-axis at Q and normal at it meet the parabola S at the point R. Then the area (in sq. units) of the triangle PQR is equal to :
[JEE Main 2021, 20 Jul (Shift 1)]
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The locus of the centroid of the triangle formed by any point P on the hyperbola 16x2 – 9y2 + 32x + 36y – 164 = 0, and its foci is
[JEE Main 2021, 25 Jul (Shift 1)]
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Let \(\mathrm{P}\left({\mathrm{x}}_{0},{\mathrm{y}}_{0}\right)\) be the point on the hyperbola \(3{x}^{2}−4{y}^{2}\) = 36 which is nearest to the line 3x + 2y = 1. Then \(\sqrt{2}\text{ }\left({y}_{0}−{x}_{0}\right)\) is equal to
[JEE Main 2023, 1 Feb (Shift 2)]
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If the curves, \(\frac{{x}^{2}}{a}+\frac{{y}^{2}}{b}=1\) and \(\frac{{x}^{2}}{c}+\frac{{y}^{2}}{d}=1\) intersect each other at an angle of \(90^\circ\), then which of the following relations is TRUE?
[JEE Main 2021, 25 Feb (Shift 1)]
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If the tangent at a point \(P\) on the parabola \(y^2=3 x\) is parallel to the line \(x+2 y=1\) and the tangents at the points \(Q\) and \(R\) on the ellipse \(\frac{x^2}{4}+\frac{y^2}{1}=1\) are perpendicular to the line \(x-y=2\), then the area of the triangle \(P Q R\) is:
[JEE Main 2023, 29 Jan (Shift 2)]
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Consider the parabola with vertex \(\left(\frac{1}{2}, \frac{3}{4}\right)\) and the directrix \(y=\frac{1}{2}\) Let \(P\) be the point where the parabola meets the line \(x=-\frac{1}{2}\) If the normal to the parabola at \(P\) intersects the parabola again at the point \(Q\), then \((P Q)^2\) is equal to:
[JEE Main 2021, 1 Sep (Shift 2)]
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The distance of the point \((6,-2 \sqrt{2})\) from the common tangent \(y=m x+c, m>0\), of the curves \(x=2 y^2\) and \(x=1+y^2\) is
[JEE Main 2023, 25 Jan (Shift 1)]
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Let \(H\) be the hyperbola, whose foci are \((1 \pm \sqrt{2}, 0)\) and eccentricity is \(\sqrt{2}\). Then the length of its latus rectum is
[JEE Main 2023, 31 Jan (Shift 2)]
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The locus of mid. points of the line segments joining \((-3,-5)\) and the points on the ellipse \(\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}=1\) is:
[JEE Main 2021, 31 Aug (Shift 2)]
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If \(P(h, k)\) be point on the parabola \(x=4 y^2\), which is nearest to the point \(Q(0,33)\), then the distance of \(P\) from the directrix of the parabola \(y^2=4(x+y)\) is equal to:
[JEE Main 2023, 30 Jan (Shift 1)]
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Let the hyperbola \(H: \frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) pass through the point \((2 \sqrt{2},-2 \sqrt{2})\). A parabola is drawn whose focus is same as the focus of \(H\) with positive abscissa and the directrix of the parabola passes through the other focus of \(H\). If the length of the latus rectum of the parabola is \(e\) times the length of the latus rectum of \(H\), where \(e\) is the eccentricity of \(H\), then which of the following points lies on the parabola?
[JEE Main 2022, 28 Jul (Shift 2)]
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If \(\frac{x}{ma}+\frac{y}{nb}=1\) touches the ellipse \(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\), then
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Let the tangent and normal at the point \((3 \sqrt{3}, 1)\) on the ellipse \(\frac{x^2}{36}+\frac{y^2}{4}=1\) meet the \(y\)-axis at the points \(A\) and \(B\) respectively. Let the circle \(C\) be drawn taking \(A B\) as a diameter and the line \(x=2 \sqrt{5}\) intersect \(C\) at the points \(P\) and \(Q\). If the tangents at the points \(P\) and \(Q\) on the circle intersect at the point \((\alpha, \beta)\), then \(\alpha^2-\beta^2\) is equal to
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If the eccentricity and length of latus rectum of a hyperbola are \(\frac{\sqrt{13}}{3}\) and \(\frac{10}{3}\) units respectively, then what is the length of the transverse axis?
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Let a line \( L: 2 x+y=k, k>0 \) be a tangent to the hyperbola \( x^{2}-y^{2}=3 \). If \( L \) is also a tangent to the parabola \( y^{2}=\alpha x \), then \( \alpha \) is equal to:
[JEE Main 2021, 22 Jul (Shift 2)]
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The eccentricity of the ellipse whose major axis is three times the minor axis is:
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Let \(P\left(x_0, y_0\right)\) be the point on the hyperbola \(3 x^2-4 y^2=36\), which is nearest to the line \(3 x+2 y=1\). Then \(\sqrt{2}\left(y_0-x_0\right)\) is equal to:
[JEE Main 2023, 1 Feb (Shift 2)]
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The eccentricity of the hyperbola \(16{x}^{2}-9{y}^{2}=1,\) is
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If the normal at an end of a latus rectum of an ellipse passes through an extremity of the minor axis, then the eccentricity \(e\) of the ellipse satisfies:
[JEE Main 2020, 6 Sep (Shift 2)]
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Let \(L_1\) be a tangent to the parabola \(y^2=4(x+1)\) and \(L_2\) be a tangent to the parabola \(y^2=8(x+2)\) such that \(L_1\) and \(L_2\) intersect at right angles. Then \(L_1\) and \(L_2\) meet on the straight line:
[JEE Main 2020, 6 Sep (Shift 1)]
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If the curve \({x}^{2}+2{y}^{2}=2\) intersects the line \(x+y=1\) at two points \(P\) and \(Q\), then the angle subtended by the line segment \(P Q\) at the origin is:
[JEE Main 2021, 25 Feb (Shift 2)]
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Let a tangent to the curve \(y^2=24 x\) meet the curve \(x y=\) 2 at the points \(A\) and \(B\). Then the mid points of such line segments \(A B\) lie on a parabola with the
[JEE Main 2023, 24 Jan (Shift 1)]
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The locus of a point which divides the line segment joining the point \( (0,-1) \) and a point on the parabola, \( x^{2}=4 y \), internally in the ratio \( 1: 2 \), is:
[JEE Main 2020, 8 Jan (Shift 1)]
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If a line along a chord of the circle \(4 x^2+4 y^2+120 x+675=0\), passes through the point \((-30,0)\) and is tangent to the parabola \(y^2=30 x\), then the length of this chord is:
[JEE Main 2021, 26 Aug (Shift 1)]
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If two tangents drawn from a point \(P\) to the parabola \(y^2=\) \(16(x-3)\) are at right angles, then the locus of point \(P\) is:
[JEE Main 2021, 27 Aug (Shift 2)]
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On the ellipse \(\frac{{x}^{2}}{8}+\frac{{y}^{2}}{4}=1\) let P be a point in the second quadrant such that the tangent at P to the ellipse is perpendicular to the line \(x+2y=0\). Let S and S ' be the foci of the ellipse and e be its eccentricity. If A is the area of the triangle SPS' then, the value of \(\left(5-{e}^{2}\right)\) . A is:
[JEE Main 2021, 26 Aug (Shift 1)]
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The locus of mid.points of the line segments joining (-3,-5) and the points on the ellipse \(\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}=1\) is:
[JEE Main 2021, 31 Aug (Shift 2)]
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The locus of mid. points of the line segments joining \((-3,-5)\) and the points on the ellipse \(\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}=1\) is:
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The point \(P(-2\sqrt{6},\sqrt{3})\) lies on the hyperbola \(\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1\) having eccentricity \(\frac{\sqrt{5}}{2}\). If the tangent and normal at\(P\) to the hyperbola intersect its conjugate axis at the points \(Q\) and \(R\) respectively, then \(QR\) is equal to:
[JEE Main 2021, 26 Aug (Shift 2)]
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For which of the following curves, the line \(x+\sqrt{3}y=2 \sqrt{3}\) is the tangent at the point \(\left(\frac{3 \sqrt{3}}{2}, \frac{1}{2}\right)\) ?
[JEE Main 2021, 24 Feb (Shift 2)]
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The length of the latus rectum of parabola, whose vertex and focus are on the positive x-axis at a distance R and S (S > R) respectively from the origin, is:
[JEE Main 2021, 31 Aug (Shift 1)]
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Let \(y=f(x)\) represent a parabola with focus \(\left(-\frac{1}{2}, 0\right)\) and directrix \(y=-\frac{1}{2}\). Then
\(S=\left\{x \in R: \tan ^{-1}\left(\sqrt{f(x)}+\sin ^{-1}(\sqrt{f(x)+1})\right)=\frac{\pi}{2}\right\}\) :
[JEE Main 2023, 31 Jan (Shift 1)]
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The locus of the midpoints of the chord of the circle, \( x^{2}+y^{2}=25 \) which is tangent to the hyperbola, \( \frac{x^{2}}{9}-\frac{y^{2}}{16}=1 \). is :
[JEE Main 2021, 16 Mar (Shift 1)]
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A hyperbola having the transverse axis of length \(\sqrt{2}\) has the same foci as that of the ellipse of \(3 x^2+4 y^2=12\), then this hyperbola does not pass through which of the following points?
[JEE Main 2020, 3 Sep (Shift 1)]
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Let \(L\) be a tangent line to the parabola \(y^2=4 x-20\) at \((6,2)\). If \(L\) is also a tangent to the ellipse \(\frac{x^2}{2}+\frac{y^2}{b}=1\), then the value of \(b\) is equal to:
[JEE Main 2021, 17 Mar (Shift 2)]
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The parabolas : \(a x^2+2 b x+c y=0\) and \(d x^2+2 e x+f y=0\) intersect on the line \(y=1\). If \(a, b, c, d, e, f\) are positive real numbers and \(a, b, c\) are in G.P., then
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Let the tangent and normal at the point \((3\sqrt{3},1)\) on the ellipse \(\frac{{x}^{2}}{36}+\frac{{y}^{2}}{4}=1\) meet the y - axis at the points A and B respectively. Let the circle C be drawn taking AB as a diameter and the line \(x=2\sqrt{5}\) intersect C at the points P and Q. If the tangents at the points P and Q on the circle intersect at the point \((\alpha ,\beta )\), then \({\alpha }^{2}-{\beta }^{2}\) is equal to
[JEE Main 2023, 13 Apr (Shift 1)]
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The ordinates of the points \(P\) and \(Q\) on the parabola with focus \((3,0)\) and directrix \(x=-3\) are in the ratio \(3: 1\). If \(R(\alpha, \beta)\) is the point of intersection of the tangents to the parabola at \(P\) and \(Q\), then \(\frac{\beta^2}{\alpha}\) equal to________
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Let \(P Q\) be a focal chord of the parabola \(y^2=36 x\) of length 100 , making an acute angle with the positive \(x\)-axis. Let the ordinate of \(P\) be positive and \(M\) be the point on the line segment \(P Q\) such that \(P M: M Q=3: 1\). Then which of the following points does NOT lie on the line passing through \(M\) and perpendicular to the line \(P Q\) ?
[JEE Main 2023, 13 Apr (Shift 1)]
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The equation of a common tangent to the parabolas \(y=x^2\) and \(y=-(x-2)^2\) is
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If the curve \({x}^{2}+2{y}^{2}=2\) intersects the line \(x+y=1\) at two points P and Q, then the angle subtended by the line segment P Q at the origin is:
[JEE Main 2021, 25 Feb (Shift 2)]
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Let \(L_1\) be a tangent to the parabola \(y^2=4(x+1)\) and \(L_2\) be a tangent to the parabola \(y^2=8(x+2)\) such that \(L_1\) and \(L_2\) intersect at right angles. Then \(L_1\) and \(L_2\) meet on the straight line:
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If the maximum distance of normal to the ellipse \(\frac{{x}^{2}}{4}+\frac{{y}^{2}}{{b}^{2}}=1,b<2\), from the origin is 1 , then the eccentricity of the ellipse is:
[JEE Main 2023, 31 Jan (Shift 1)]
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If the line \(y = mx + c\) is a common tangent to the hyperbola \(\frac{x^{2}}{100} - \frac{y^{2}}{64} = 1\ \) and the circle \(x^{2} + y^{2} = 36,\ \)then which one of the following is true ?
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Let \(A\) be a point on the \(x\)-axis. Common tangents are drawn from \(A\) to the curves \(x^2+y^2=8\) and \(y^2=16 x\). If one of these tangents touches the two curves at \(Q\) and \(R\), then \((Q R)^2\) is equal to
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If \(\frac{x}{\mathrm{ma}}+\frac{\mathrm{y}}{\mathrm{nb}}=1\) touches the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\), then
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The length of the latus rectum of the parabola which has focus at (\(-\)1, 1) and the directrix is \(4x+3y-24=0\) is
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Let \(\theta\) be the acute angle between the tangents to the ellipse \(\frac{x^2}{9}+\frac{y^2}{1}=1\) and the circle \(x^2+y^2=3\) at their point of intersection in the first quadrant. The \(\tan \theta\) is equal to:
[JEE Main 2021, 1 Sep (Shift 2)]
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The line \(12 x \cos \theta+5 y \sin \theta=60\) is tangent to which of the following curves?
[JEE Main 2021, 31 Aug (Shift 1)]
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If the normal at an end of a latus rectum of an ellipse passes through an extremity of the minor axis, then the eccentricity e of the ellipse satisfies:
[JEE Main 2020, 6 Sep (Shift 2)]
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The locus of the mid points of the chords of the hyperbola \({x}^{2}-{y}^{2}=4\), which touch the parabola \({y}^{2}=8x\), is:
[JEE Main 2021, 26 Aug (Shift 2)]
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