JEE Mathematics 2023 Question with Solution

Given: The adjacent sides are $2x - 3y = -23$ and $5x + 4y = 23$, and one diagonal is $3x + 7y = 23$.

Find: The value of $50d^2$, where $d$ is the distance of point $A$ from the other diagonal.

First, find the vertex $A$ as the intersection of the two given side lines:

Solving gives $A(-4,5)$.

Using the diagonal $AC$, the other endpoint is obtained as $C(3,2)$.

Now find the midpoint of diagonal $AC$:

The diagonals of a parallelogram bisect each other, so diagonal $BD$ also passes through $\left( -\frac{1}{2}, \frac{7}{2} \right)$. Using the line through the points $B(-1,-7)$ and $D(1,2)$, the equation of diagonal $BD$ is obtained as:

Now use the distance formula from point $A(-4,5)$ to the line $7x + y = 0$:

Therefore,

So, the correct option is A.