Circles
39 JEE Maths previous year questions on Circles — options free on every question; 4 include the answer & explanation free, the rest unlock with PYQ Pass.
Four distinct points \((2 k, 3 k),(1,0),(0,1)\) and \((0,0)\) lie on a circle for \(k\) equal to
[JEE Main 2024, 27 Jan (Shift 1)]
\(\frac{5}{13}\)
Given four points \((2 k, 3 k),(1,0),(0,1),(0,0)\)
since \((1,0)\) and \((0,1)\) subtends an angle of \(90^{\circ}\) at \((0,0)\)
Circle passes through \((1,0),(0,1),(0,0)\)
\((x-1)(x-0)+(y-0)(y-1)=0\)
\( x^2+y^2-x-y=0 \)
\((2 k, 3 k)\) lies on circle.
\( \Rightarrow 4 k^2+9 k^2-2 k-3 k=0 \)
\( k=\frac{5}{13}\)
Four distinct points \((2 k, 3 k),(1,0),(0,1)\) and \((0,0)\) lie on a circle for \(k\) equal to
[JEE Main 2024, 27 Jan (Shift 1)]
\(\frac{5}{13}\)
Given four points \((2 k, 3 k),(1,0),(0,1),(0,0)\)
since \((1,0)\) and \((0,1)\) subtends an angle of \(90^{\circ}\) at \((0,0)\)
Circle passes through \((1,0),(0,1),(0,0)\)
\((x-1)(x-0)+(y-0)(y-1)=0\)
\( x^2+y^2-x-y=0 \)
\((2 k, 3 k)\) lies on circle.
\( \Rightarrow 4 k^2+9 k^2-2 k-3 k=0 \)
\( k=\frac{5}{13}\)
Let a circle pass through the origin and its centre be the point of intersection of two mutually perpendicular lines \(x+(k-1)y+3=0\) and \(2x+{k}^{2}y-4=0\). If the line \(x-y+2=0\) intersects the circle at the points \(A\) and \(B,\) then \((AB{)}^{2}\) is equal to:
[JEE Main 2026, 2 Apr (Shift 2)]
\(18\)
\( x+(k-1) y+3=0 \)
\( 2 x+k^2 y-4=0\)
Since these \(2\) lines are perpendicular
\( \frac{1}{1-k}\left(-\frac{2}{k^2}\right)=-1 \)
\( 2=k^2(1-k) \)
\( \Rightarrow k^3-k^2+2=0\)
\(k^3-k^2+2=(k+1)\left(k^2-2 k+2\right)=0\),
and since \(k\) is real, \(k=-1\)
\(∴\) Lines are :
\(x-2 y+3=0 \) and \(2 x+y-4=0\)
then \( x=1\) and \(y=2\)
\(∴\) Centre is \((1,2)\)
Circle will be \(x^2+y^2-2 x-4 y=0\)
Line \(x-y+2=0\) will intersect at \(A(-1,1)\) and \(B(2,4)\)
\( \therefore(A B)^2=(2+1)^2+(4-1)^2 \)
\( =9+9=18\)
Let \(y=x\) be the equation of a chord of the circle \({C}_{1}\) (in the closed half-plane\(x\geq 0\)) of diameter \(10\) passing through the origin. Let \({C}_{2}\) be another circle described on the given chord as its diameter. If the equation of the chord of the circle \({C}_{2}\), which passes through the point \((2,3)\) and is farthest from the center of \({C}_{2},\) is \(x+ay+b=0\), then \(a–b\) is equal to
[JEE Main 2026, 28 Jan (Shift 1)]
\(–2\)
The circle \(C_1\) has a diameter of \(10 \), so its radius is \(R=5\). It passes through the origin \((0,0)\) and is contained in the closed half-plane \(x \geq 0\).
Therefore centre is \((5,0)\)
The equation of \(C_1\) is:
\((x-5)^2+y^2=25\)
\(\Rightarrow x^2+y^2-10 x=0\)
The line \(y=x\) is a chord of \(C_1\). We find the intersection points of \(y=x\) and \(C_1\) :
\((x-5)^2+x^2=25 \)
\( \Rightarrow x^2-10 x+25+x^2=25 \)
\( \Rightarrow 2 x^2-10 x=0 \)
\( \Rightarrow 2 x(x-5)=0\)
The intersection points are \((0,0)\) and \((5,5)\). These are the endpoints of the diameter of circle \(C_2\).
The center \(M\) of \(C_2\) is the midpoint of the chord:
\(M\left(\frac{5}{2}, \frac{5}{2}\right)\)
Given point is \(P(2,3)\)
In any circle, the chord passing through a point \(P\) that is farthest from the center is the one perpendicular to the radius (or segment) \(M P\).
Now the slope of the segment \(M P\) is:
\(m_{M P}=\frac{3-2.5}{2-2.5}=\frac{0.5}{-0.5}=-1\)
The slope \(m\) of the required chord is
\(m=-\frac{1}{-1}=1\)
Now equation of required chord is \(y-3=1(x-2)\)
\(\Rightarrow y-3=x-2 \Rightarrow x-y+1=0\)
we get \( a=-1\) and \( b=1\)
Now \(a-b=-1-1=-2\)
Circle lie in the second quadrant with radius 2 and touching both coordinate axes. Another circle with centres \((2,6)\) exactly intersect the first circle at two points then range of it's radius is \((\mathrm{a}, \mathrm{b})\) then find \((\mathrm{a}+\mathrm{b})\).
Options are free to see. Unlock the correct answer and full explanation with Pass.
Circle lie in the second quadrant with radius 2 and touching both coordinate axes. Another circle with centres \((2,6)\) exactly intersect the first circle at two points then range of it's radius is \((\mathrm{a}, \mathrm{b})\) then find \((\mathrm{a}+\mathrm{b})\). (22 Jan, Shift I, Memory Based)
Options are free to see. Unlock the correct answer and full explanation with Pass.
If the points of intersection of the ellipses \({x}^{2}+2{y}^{2}−6x−12y+23=0\) and \(4{x}^{2}+2{y}^{2}−20x−12y+35=0\) lie on a circle of radius \(r\) and centre \((a,b),\) then the value of \(ab+18{r}^{2}\) is
[JEE Main 2026, 23 Jan (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
If the four distinct points \((4,6),(-1,5),(0,0)\)and \((\mathrm{k},3\mathrm{k})\) lie on a circle of radius \(r\), then \(10\mathrm{k}+{\mathrm{r}}^{2}\) is equal to
[JEE Main 2025, 3 Apr (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
If the four distinct points \((4,6),(-1,5),(0,0)\)and \((\mathrm{k},3\mathrm{k})\) lie on a circle of radius \(r\), then \(10\mathrm{k}+{\mathrm{r}}^{2}\) is equal to
Options are free to see. Unlock the correct answer and full explanation with Pass.
\({x}^{2}+{y}^{2}-\alpha x-\beta y+\gamma =0\), this circle touches \(x\) axis at \((a,0)\) and it's \(Y\) intercept is \(b\). Find \(\left(2a,{b}^{2}\right)\) in terms of \(\alpha ,\beta ,\gamma\). (28 Jan, Shift I, Memory Based)
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let the circle \({x}^{2}+{y}^{2}=4\) intersect \(x\)-axis at the points \(A(a,0),a>0\) and \(B(b,0)\). Let \(P(2\cos \alpha ,2\sin \alpha ),\)and \(Q(2\cos \beta ,2\sin \beta )\) be two points such that \(\left(\alpha −\beta \right)=\frac{\pi }{2}.\) Then the point of intersection of \(AQ\) and \(BP\) lies on:
[JEE Main 2026, 28 Jan (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
A circle \(C\) of radius \(2\) lies in the second quadrant and touches both the coordinate axes. Let \(r\) be the radius of a circle that has centre at the point \((2, 5)\) and intersects the circle \(C\) at exactly two points. If the set of all possible values of \(r\) is the interval \((\alpha ,\beta )\) then \(3\beta -2\alpha\) is equal to :
[JEE Main 2025, 22 Jan (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
\({x}^{2}+{y}^{2}-\alpha x-\beta y+\gamma =0\), this circle touches \(x\) axis at \((a,0)\) and it's \(Y\) intercept is \(b\). Find \(\left(2a,{b}^{2}\right)\) in terms of \(\alpha ,\beta ,\gamma\). (28 Jan, Shift I, Memory Based)
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let the circle \(S: 36 x^{2}+36 y^{2}-108 x+120 y+C=0\) be such that it neither intersects nor touches the co-ordinate axes. If the point of intersection of the lines, \(x-2 y=4\) and \(2 x-y=5\) lies inside the circle \(S\), then :
[JEE Main 2021, 22 Jul (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Two tangents are drawn from a point \(P\) to the circle \(x^2+y^2-2 x-4 y+4=0\), such that the angle between these tangents is \(\tan ^{-1}\left(\frac{12}{5}\right)\) where \(\tan ^{-1}\left(\frac{12}{5}\right) \in(0, \pi)\) If the centre of the circle is denoted by \(C\) and these tangents touch the circle at points \(A\) and \(B\), then the ratio of the areas of \(\triangle P A B\) and \(\triangle C A B\) is:
[JEE Main 2021, 17 Mar (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let the lengths of intercepts on \(x\)-axis and \(y\)-axis made by the circle \(x^2+y^2+a x+2 a y+c=0,(a<0)\) be \(2 \sqrt{2}\) and \(2 \sqrt{5}\), respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line \(x+2 y=0\), is equal to :
[JEE Main 2021, 16 Mar (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
The number of common tangents, to the circles \(x^2+y^2\) \(-18 x-15 y+131=0\) and \(x^2+y^2-6 x-6 y-7=0\), is:
[JEE Main 2023, 15 Apr (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
The line \(2 x-y+1=0\) is a tangent to the circle at the point \((2,5)\) and the centre of the circle lies on \(x-2 y=4\). Then, the radius of the circle is:
[JEE Main 2021, 17 Mar (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
If the coordinates at one end of a diameter of the circle \(x^2+y^2-8 x-4 y+c=0\) are \((-3,2)\), then the coordinates at the other end are
Options are free to see. Unlock the correct answer and full explanation with Pass.
A circle with centre \((2,3)\) and radius 4 intersects the line \(x+y=3\) at the points \(P\) and \(Q\). If the tangents at \(P\) and \(Q\) intersect at the point \(S(\alpha, \beta)\), then \(4 \alpha-7 \beta\) is equal to
[JEE Main 2023, 29 Jan (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Consider a circle \(C_1: x^2+y^2-4 x-2 y=\alpha-5\). Let its mirror image in the line \(y=2 x+1\) be another circle \(C_2: 5 x^2+5 y^2-10 f x-10 g y+36=0\). Let \(r\) be the radius of \(C_2\). Then \(\alpha+r\) is equal to ________
[JEE Main 2023, 8 Apr (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
. Let the tangents at the points \(A(4,-11)\) and \(B(8,-5)\) on the circle \(x^{2}+y^{2}-3 x+10 y-15=0\), intersect at the point \(C\). Then the radius of the circle, whose centre is \(C\) and the line joining \(A\) and \(B\) is its tangent, is equal to
[JEE Main 2023, 29 Jan (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
A circle C touches the line \(x=2y\) at the point (2,1) and intersects the circle \({C}_{1}:{x}^{2}+{y}^{2}+2y-5=\) 0 at two points P and Q such that PQ is a diameter of \({C}_{1}\). Then the diameter of C is
[JEE Main 2021, 26 Aug (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
A line segment \(A B\) of length \(\lambda\) moves such that the points \(A\) and \(B\) remain on the periphery of a circle of radius \(\lambda\). Then the locus of the point, that divides the line segment \(A B\) in the ratio \(2: 3\), is circle of radius
[JEE Main 2023, 10 Apr (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let the tangents at the points \(A(4,-11)\) and \(B(8,-5)\) on the circle \(x^2+y^2-3 x+10 y-15=0\), intersect at the point \(C\). Then the radius of the circle, whose centre is \(C\) and the line joining \(A\) and \(B\) is its tangent, is equal to
[JEE Main 2023, 29 Jan (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
A circle with centre \((2,3)\) and radius 4 intersects the line \(x+y=3\) at the points \(P\) and \(Q\). If the tangents at \(P\) and \(Q\) intersect at the point \(S(\alpha, \beta)\), then \(4 \alpha-7 \beta\) is equal to__________
Options are free to see. Unlock the correct answer and full explanation with Pass.
A circle \(C\) touches the line \(x=2 y\) at the point \((2,1)\) and intersects the circle \(C_1: x^2+y^2+2 y-5=0\) at two points \(P\) and \(Q\) such that \(P Q\) is a diameter of \(C_1\). Then the diameter of \(C\) is
[JEE Main 2021, 26 Aug (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let the lengths of intercepts on \(x\)-axis and \(y\)-axis made by the circle \(x^2+y^2+a x+2 a y+c=0,(a<0)\) be \(2 \sqrt{2}\) and \(2 \sqrt{5}\), respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line \(x+2 y=0\), is equal to :
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \(S_1: x^2+y^2=9\) and \(S _2:(x-2)^2+y^2=1\). Then the locus of center of a variable circle \(S\) which touches \(S_1\) internally and \(S_2\) externally always passes through the points:
[JEE Main 2021, 18 Mar (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let the lengths of intercepts on x–axis and y-axis made by the circle x2 + y2 + ax + 2ay + c = 0,
(a < 0) be \(2\sqrt{2}\) and \(2\sqrt{5},\) respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line x + 2y = 0, is equal to
[JEE Main 2021, 16 Mar (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
A circle \(C\) touches the line \(x=2 y\) at the point \((2,1)\) and intersects the circle \(C_1: x^2+y^2+2 y-5=0\) at two points \(P\) and \(Q\) such that \(P Q\) is a diameter of \(C_1\). Then the diameter of \(C\) is
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \(A\) be the point \((1,2)\) and \(B\) be any point on the curve \(x^2+y^2=16\). If the centre of the locus of the point \(P\), which divides the line segment \(A B\) in the ratio \(3: 2\) is the point \(C\) \((\alpha, \beta)\), then the length of the line segment \(A C\) is
[JEE Main 2023, 10 Apr (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Choose the incorrect statement about the two circles whose equations are given below :
\(x^2+y^2-10 x-10 y+41=0\) and \(x^2+y^2-16 x-10 y+80=0 \)
[JEE Main 2021, 17 Mar (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
If a chord of the circle \(x^2+y^2=8\) makes equal intercets of length \(a\) on the coordinate axes, then
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let
\(A=\left\{(x,y)\in R\times R∣2{x}^{2}+2{y}^{2}-2x-2y=1\right\}\\ B=\left\{(x,y)\in R\times R∣4{x}^{2}+4{y}^{2}-16y+7=0\right\}\text{ and }\\ C=\left\{(x,y)\in R\times R∣{x}^{2}+{y}^{2}-4x-2y+5\leq {r}^{2}\right\}\)
Then the minimum value of \(|r|\) such that \(A\cup B\subseteq C\) is equal to :
[JEE Main 2021, 27 Jul (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
If a chord of the circle \({x}^{2}+{y}^{2}=8\) makes equal intercepts of length a on the coordinate axes, then
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \(B\) be the centre of the circle \(x^{2} + y^{2} - 2x + 4y + 1 = 0\). Let the tangents at two points \(P\) and \(Q\) on the circle intersect at the point \(A(3,1)\). Then \(8.\left( \frac{\text{area } \bigtriangleup \text{APQ}}{\text{area } \bigtriangleup \text{BPQ}} \right)\) is equal to
[JEE Main 2021, 31 Aug (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
The number of common tangents, to the circles \(x^{2}+y^{2}\) \(-18 x-15 y+131=0\) and \(x^{2}+y^{2}-6 x-6 y-7=0\), is:
[JEE Main 2023, 15 Apr (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Choose the correct statement about two circles whose equations are given below: \(x^{2}+y^{2}-10 x-10 y+41=0\), \(x^{2}+y^{2}-22 x-10 y+137=0\)
[JEE Main 2021, 18 Mar (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Practice more JEE Maths PYQs
Browse every Maths chapter, or explore the full JEE question bank.
All Maths chapters →