Mathematics · Straight lines

JEE Main 2024 — 9 April, Shift 1 — Question 4

A ray of light coming from the point $P(1,2)$ gets reflected from the point Q on the x -axis and then passes through the point $R(4,3)$ . If the point $S(h$ , k ) is such that PQRS is a parallelogram, then $h \mathrm{k}^{2}$ is equal to :

Option A:
80
Option B:
90
Option C:
60
Option D:
70
Correct

Answer: D

Step-by-step solution

Image of P wrt x -axis will be $\mathrm{P}^{\prime}(1,-2)$

equation of line joining P'R will be $y-3=\frac{5}{3}(x-4)$

Above line will meet x -axis at Q where $\mathrm{y}=0 \Rightarrow \mathrm{x}=\frac{11}{5}$

$\therefore \mathrm{Q}\left(\frac{11}{5}, 0\right)$

$\because$ PQRS is parallelogram so their diagonals will bisects each other

$\Rightarrow \frac{4+1}{2}=\frac{\frac{11}{5}+\mathrm{h}}{2} \& \frac{2+3}{2}=\frac{\mathrm{k}+0}{2}$

$\Rightarrow \mathrm{h}=\frac{14}{5} \& \mathrm{k}=5$

$\therefore \mathrm{hk}^{2}=\frac{14}{5} \times 5^{2}=70$

Answer key and solution verified before publishing.

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Exam: JEE Main 2024
Paper: 9 April, Shift 1
Subject: Mathematics
Chapter: Straight lines
Topic: Angle between lines, perpendicular distance & distance between parallel lines, foot, image