JEE Mathematics 2024 Question with Solution

Given: A ray starts from $P(1,2)$, reflects at $Q$ on the $x$-axis, and passes through $R(4,3)$. Also, $PQRS$ is a parallelogram.

Find: The value of $hk^2$ where $S(h,k)$.

Let

since $Q$ lies on the $x$-axis.

Using the law of reflection at the $x$-axis, the slopes of $PQ$ and $QR$ are negatives of each other:

Now solve for $a$:

Hence,

Since $PQRS$ is a parallelogram, the diagonals bisect each other. So midpoint of $PR$ equals midpoint of $QS$.

Midpoint of $PR$ is

If

then midpoint of $QS$ is

Equating the midpoints:

So,

and

Now compute:

Therefore, the value of $hk^2$ is $70$. The correct option is D.

Given: $P(1,2)$, $Q$ lies on the $x$-axis, and the reflected ray passes through $R(4,3)$.

Find: $hk^2$ for $S(h,k)$ such that $PQRS$ is a parallelogram.

Let

from the reflection condition obtained by equating the slopes with opposite signs.

Now use the parallelogram property that

Compute

Therefore,

So,

Hence,

Therefore, the required value is $70$, so the correct option is D.