Straight Lines

Let the range of the function \(f(x)=6+16 \cos x \cdot \cos \left(\frac{\pi}{3}-x\right) \cdot \cos \left(\frac{\pi}{3}+x\right)\)\(\sin 3 x \cdot \cos 6 x, x \in R\) be \([\alpha, \beta]\). Then the distance of the point \((\alpha, \beta)\) from the line \(3 x+4 y+12=0\) is :

[JEE Main 2025, 23 Jan (Shift 2)]

Consider the lines \(\mathrm{x}(3\lambda +1)+\mathrm{y}(7\lambda +2)=17\lambda +5\), \(\lambda\) being a parameter, all passing through a point \(P\). One of these lines (say \(L\)) is farthest from the origin. If the distance of \(L\)from the point\((3,6)\) is \(d\), then the value of \({d}^{2}\) is

[JEE Main 2025, 3 Apr (Shift 2)]

If the images of the points \(\mathbf{A}(1,3), \mathbf{B}(3,1)\) and \(\mathbf{C}(2,4)\) in the line \(x+2 y=4\) are D, E and F respectively, then the centroid of the triangle DEF is

If the locus of the point, whose distances from the points \((2,1)\) and \((1,3)\) are in the ratio \(5: 4\), is \(a x^2+b y^2+c x y+d x+e y+170=0\), then the value of \(a^2+2 b+3 c+4 d+e\) is equal to:

[JEE Main 2024, 6 Apr (Shift 2)]

Let the triangle \(P Q R\) be the image of the triangle with vertices \((1,3),(3,1)\) and \((2,4)\) in the line \(x+2 y=2\). If the centroid of \(\Delta PQR\) is the point \((\alpha, \beta)\), then \(15(\alpha-\beta)\) is equal to :

[JEE Main 2025, 22 Jan (Shift 1)]

A triangle \(A B C\) have sides \(x+y=11 ; 2 x+3 y=29\) and \(x+2y=16\) such that a point \(P\left(\frac{11}{2}, \alpha\right)\) lies on or inside the triangle find the product of maximum and minimum value of " \(\alpha\) ".

Let \({}^{n}C_{r-1}=28,{}^{n}C_{r}=56\text{ and }{}^{n}C_{r+1}=70\). Let \(\mathrm{A}(4\cos t,4\sin t),\mathrm{B}(2\mathrm{sint},-2\mathrm{cost})\)\(\text{ and }C\left(3r-n,{r}^{2}-n-1\right)\) be the vertices of a triangle \(ABC\), where \(t\) is a parameter. If \((3x-1{)}^{2}+(3y{)}^{2}=\alpha ,\) is the locus of the centroid of triangle \(ABC\), then\(\alpha\)equals:

[JEE Main 2025, 28 Jan (Shift 1)]

Let the line \({L}_{1}:x+3=0\) intersect the lines \({L}_{2}:x-y=0\) and \({L}_{3}:3x+y=0\) at the points \(A\) and \(B,\) respectively. Let the bisector of the obtuse angle between the lines \({L}_{2}\text{and}{L}_{3}\) intersect the line \({L}_{1}\) at the point \(C.\) Then \(B{C}^{2}:A{C}^{2}\) is equal to:

[JEE Main 2026, 4 Apr (Shift 1)]

Two sides of isoceles triangle are \(x+2y=4\) and \(x+y=4\). Then the sum of all possible values of \(m\) where \(m\) is slope of 3rd side,is

A triangle \(A B C\) have sides \(x+y=11 ; 2 x+3 y=29\) and \(x+2y=16\) such that a point \(P\left(\frac{11}{2}, \alpha\right)\) lies on or inside the triangle find the product of maximum and minimum value of " \(\alpha\) ".

The distance from the origin to the image of \((1,1)\) with respect to the line \(x+y+5=0\) is:

Two sides of isoceles triangle are \(x+2y=4\) and \(x+y=4\). Then the sum of all possible values of \(m\) where \(m\) is slope of 3rd side,is

The distance from the origin to the image of \((1,1)\) with respect to the line \(x+y+5=0\) is:

If the images of the points \(\mathbf{A}(1,3), \mathbf{B}(3,1)\) and \(\mathbf{C}(2,4)\) in the line \(x+2 y=4\) are D, E and F respectively, then the centroid of the triangle DEF is (24 Jan, Shift I, Memory Based)

Let the triangle \(P Q R\) be the image of the triangle with vertices \((1,3),(3,1)\) and \((2,4)\) in the line \(x+2 y=2\). If the centroid of ΔPQR is the point \((\alpha, \beta)\), then \(15(\alpha-\beta)\) is equal to :

Let the range of the function

\(f(x)=6+16 \cos x \cdot \cos \left(\frac{\pi}{3}-x\right) \cdot \cos \left(\frac{\pi}{3}+x\right)\)\(\sin 3 x \cdot \cos 6 x, x \in R\)

be \([\alpha, \beta]\). Then the distance of the point \((\alpha, \beta)\) from the line \(3 x+4 y+12=0\) is :

[JEE Main 2025, 23 Jan (Shift 2)]

Let the lines \(3x-4y-\alpha =0,8x-11y-33=0\) and \(2x-3y+\lambda =0\) be concurrent. If the image of the point \((1, 2)\) in the line \(2x-3y+\lambda =0\) is \(\left(\frac{57}{13}, \frac{-40}{13}\right)\), then \(|\alpha \lambda|\) is equal to:

[JEE Main 2025, 24 Jan (Shift 1)]

Let \(a\) be the length of a side of a square OABC with O being the origin. Its side OA makes an acute angle \(\alpha\) with the positive x-axis and the equations of its diagonals are \((\sqrt{3}+1)x+(\sqrt{3}-1)y=0\) and \((\sqrt{3}-1)x-(\sqrt{3}+1)y+8\sqrt{3}=0\). Then \({a}^{2}\) is equal to

[JEE Main 2025, 8 Apr (Shift 1)]

A line passing through the point \(\mathrm{P}(\mathrm{a},0)\)makes an acute angle \(\alpha\) with the positive x -axis. Let this line be rotated about the point P through an angle \(\frac{\alpha }{2}\) in the clock-wise direction. If in the new position, the slope of the line is \(2-\sqrt{3}\) and its distance from the origin is \(\frac{1}{\sqrt{2}}\), then the value of \(3{\mathrm{a}}^{2}{\tan }^{2}\alpha -2\sqrt{3}\) is

[JEE Main 2025, 8 Apr (Shift 1)]

The image of a point \(A(3,8)\) in the line \(x+3y-7=0\), is

let the equation \(x(x+2)(12-k)=2\) have equal roots. Then the distance of the point \(\left(k,\frac{k}{2}\right)\) from the line \(3x+4y+5=0\) is

[JEE Main 2025, 3 Apr (Shift 2)]

If \(p\) and \(q\) are the length of the perpendiculars from the origin on the lines \(\text{xcosec}\alpha - ysec\alpha = k\ cot2\alpha\) and \(\text{xsin}\alpha + ycos\alpha = ksin2\alpha\) respectively, then \(k^{2}\) is equal to?

[JEE Main 2021, 31 Aug (Shift 1)]

What is the distance between the straight lines \(3x+4y=9\) and \(6x+8y=15?\)

Let \(A(a, 0), B(b, 2 b+1)\) and \(C(0, b), b \neq 0,|b| \neq 1\), be points such that the area of \(\triangle ABC\) is 1sq. unit, then the sum of all possible values of a is

[JEE Main 2021, 27 Aug (Shift 2)]

If the coordinates of the points A and B be (3, 3) and (7, 6), then the length of the portion of the line AB intercepted between the axes is

Let \((\alpha, \beta)\) be the centroid of the triangle formed by the lines \(15 x-y=82,6 x-5 y=-4\) and \(9 x+4 y=17\). Then \(\alpha\) \(+2 \beta\) and \(2 \alpha-\beta\) are the roots of the equation

[JEE Main 2023, 13 Apr (Shift 2)]

What is the angle between the two straight lines \(y=(2-\sqrt{3}) x+5\) and \(y=(2+\sqrt{3}) x-7 ?\)

Let \(A(a, 0), B(b, 2 b+1)\) and \(C(0, b), b \neq 0,|b| \neq 1\), be points such that the area of \(\triangle ABC\) is 1sq. unit, then the sum of all possible values of \(a\) is

[JEE Main 2021, 27 Aug (Shift 2)]

The combined equation of the two lines \(a x+b y+c=0\) and \(a^{\prime} x+b^{\prime} y+c^{\prime}=0\) can be written as \((a x+b y+c)\left(a^{\prime} x+\right.\) \(\left.b^{\prime} y+c^{\prime}\right)=0\) .The equation of the angle bisectors of the lines represented by the equation \(2 x^2+x y-3 y^2=0\) is

[JEE Main 2023, 1 Feb (Shift 1)]

The points \(\ (0,7,10),(-1,6,6) \) and \(\ (-4,9,6) \) form

A light ray emits from the origin making an angle \(30^{\circ}\) with the positive \(x\)-axis. After getting reflected by the line \(x+y=1\), if this ray intersects \(x\)-axis at \(Q\), then the abscissa of \(Q\) is

[JEE Main 2023, 29 Jan (Shift 1)]

The set of all possible values of \(\theta\) in the interval \((0, \pi)\) for which the points \((1,2)\) and \((\sin \theta, \cos \theta)\) lie on the same side of the line \(x+y=1\) is:

[JEE Main 2020, 2 Sep (Shift 2)]

Statement-1 : If the general equation \(x^2+y^2+2 x y+2 g x+2 f y+4=0\) represents a pair of real lines then \(|g| \geq 2\).
Statement-2 : The equation \(\mathrm{ax}^2+2 \mathrm{hxy}+\mathrm{by}^2+2 \mathrm{gx}+2 \mathrm{fy}+\mathrm{c}=0\) represents pair of real lines if \(abc +2 \mathrm{fgh}-\mathrm{af}^2-\mathrm{bg}^2-\mathrm{ch}^2=0\).

A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the coordinate axis is \( \frac{1}{4} \). Three stones, \( A, B \) and \( C \) are placed at the point \( (1,1),(2,2) \) and \( (4,4) \) respectively. Then which of these stones is/are on the path of the man

[JEE Main 2021, 24 Feb (Shift 1)]

Let \(a, b, c\) be in arithmetic progression. Let the centroid of the triangle with vertices \((a, c),(2, b)\) and \((a, b)\) be \(\left(\frac{10}{3}, \frac{7}{3}\right)\). If \(\alpha, \beta\) are the roots of the equation \(a x^2+b x+1=0\), then the value of \(\alpha^2+\beta^2-\alpha \beta\) is:

[JEE Main 2021, 24 Feb (Shift 2)]

If \(p\) and \(q\) are the length of the perpendiculars from the origin on the lines \(x \operatorname{cosec} \alpha-y \sec \alpha=k \cot 2 \alpha\) and \(x \sin \alpha+y \cos \alpha=k \sin 2 \alpha\) respectively, then \(k^2\) is equal to?

[JEE Main 2021, 31 Aug (Shift 1)]

The distance between the parallel lines \(\ 3 x-4 y+7=0 \) and \(\ 3 x-4 y+5=0 \) is \(\ \frac{a}{b} \). Value of \(\ a+b\) is

Consider a triangle Δ whose two sides lie on the x-axis and the line x + y + 1 = 0. If the orthocentre of Δ is (1, 1), then the equation of the circle passing through the vertices of the triangle Δ is

If \(\mathrm{p}\) and \(\mathrm{q}\) are the lengths of the perpendiculars from the origin on the lines.

\(x \operatorname{cosec} \alpha-y \sec \alpha=k \cot 2 \alpha\) and \(x \sin \alpha+y \cos \alpha=k \sin\) \(2 \alpha\) respectively, then \(k^{2}\) is equal to:

[JEE Main 2021, 31 Aug (Shift 1)]

Let the equation of the pair of lines, \(y=p x\) and \(y=q x\), can be written as \((y-p x)(y-q x)=0\). Then the equation of the pair of the angle bisectors of the lines \(x^{2}-4 x y-5 y^{2}=0\) is:

[JEE Main 2021, 25 Jul (Shift 2)]

The number of integral values of \( m \) so that the abscissa of point of intersection of lines \( 3 x+4 y \) \( =9 \) and \( y=m x+1 \) is also an integer, is :

[JEE Main 2021, 18 Mar (Shift 1)]

The equation of one of the straight lines which passes through the point \((1,3)\) and makes an angle \(\tan ^{-1}(\sqrt{2})\) with the straight line \(y+1=3 \sqrt{2} x\) is

If the perpendicular bisector of the line segment joining the points \(\mathrm{P}(1,4)\) and \(\mathrm{Q}(\mathrm{k},3)\) has \(y\)-intercept equal to \(-4,\) then a value of \(k\) is :

[JEE Main 2020, 4 Sep (Shift 2)]

If the system of equations
\[\begin{aligned}& x+2 y+3 z=3 \\& 4 x+3 y-4 z=4 \\& 8 x+4 y-\lambda z=9+\mu\end{aligned}\]
has infinitely many solutions, then the ordered pair \((\lambda, \mu)\) is equal to