JEE MAIN Straight Line Previous Year Questions (PYQs) – Page 6 of 13

If a straight line passing through the point $P(-3,4)$ is such that its intercepted portion between the coordinate axes is bisected at $P$, then its equation is:

Let $P(\alpha,\beta,\gamma)$ be the image of the point $Q(3,-3,1)$ in the line $\dfrac{x-0}{1}=\dfrac{y-3}{1}=\dfrac{z-1}{-1}$ and let $R$ be the point $(2,5,-1)$. If the area of $\triangle PQR$ is $\lambda$ and $\lambda^{2}=14K$, then $K$ is:

If the angle between the lines $\dfrac{x}{2}=\dfrac{y}{2}=\dfrac{z}{1}$ and $\dfrac{5-x}{-2}=\dfrac{7y-14}{p}=\dfrac{z-3}{4}$ is $\cos^{-1}\left(\dfrac{2}{3}\right)$, then $p$ is equal to :

Let R be a rectangle given by the lines x = 0, x = 2, y = 0 and y = 5. Let $A(\alpha, 0)$ and $B(0, \beta)$, $\alpha \in [0, 2]$ and $\beta \in [0, 5]$, be such that the line segment AB divides the area of the rectangle R in the ratio 4 : 1. Then, the mid-point of AB lies on a:

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 :

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

If the image of the point $P(a, 2, a)$ in the line $\dfrac{x}{2} = \dfrac{y + a}{1} = \dfrac{z}{1}$ is $Q$ and the image of $Q$ in the line $\dfrac{x - 2b}{2} = \dfrac{y - a}{1} = \dfrac{z + 2b}{-5}$ is $P$, then $a + b$ is equal to ______.

If the foot of the perpendicular from point (4, 3, 8) on the line ${L_1}:{{x - a} \over l} = {{y - 2} \over 3} = {{z - b} \over 4}$, l $\ne$ 0 is (3, 5, 7), then the shortest distance between the line L₁ and line ${L_2}:{{x - 2} \over 3} = {{y - 4} \over 4} = {{z - 5} \over 5}$ is equal to :

Let $729, 81, 9, 1, \ldots$ be a sequence and $P_n$ denote the product of the first $n$ terms of this sequence. If $2\sum_{n=1}^{40}(P_n)=\dfrac{3^\alpha-1}{3^\beta}$ and $\gcd(\alpha,\beta)=1$, then $\alpha+\beta$ is equal to:

A line passing through the point \(A(9,0)\) makes an angle of \(30^\circ\) with the positive direction of the \(x\)-axis. If this line is rotated about \(A\) through an angle of \(15^\circ\) in the clockwise direction, then its equation in the new position is: C