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

JEE Main · 2024 · 09 Apr 2024 · Shift 1 · Straight Lines

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

hk^{2}

is equal to :

(A)80
(B)90
(C)60
(D)70✓ Correct

Answer: (D)

Solution

Reflect

P

in the x-axis:

P^{'} = (1, - 2)

. Line

P^{'} R

has slope

\frac{3 - ( - 2 )}{4 - 1} = \frac{5}{3}

; it meets the x-axis at

Q = (\frac{11}{5}, 0)

, the reflection point. In parallelogram

P QR S

the diagonals

P R

and

QS

share a midpoint:

\frac{1}{2} (\frac{11}{5} + h) = \frac{5}{2}

and

\frac{k}{2} = \frac{5}{2}

, giving

h = \frac{14}{5}, k = 5

. So

h k^{2} = \frac{14}{5} \cdot 25 = 70

— option (4).

JEE Main · 2024 · 09 Apr 2024 · Shift 1 · Single correct option

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