JEE Mathematics 2025 Question with Solution

Given: The triangle is formed by the lines $7x - 6y + 3 = 0$, $x + 2y - 31 = 0$, and $9x - 2y - 19 = 0$.

Find: If the centroid is reflected in the line $3x + 6y - 53 = 0$, find $h^2 + k^2 + hk$.

From the solution working, the three vertices are obtained by solving pairwise intersections of the given lines:

Hence the centroid of $\triangle ABC$ is

To reflect a point $(x,y)$ in the line $ax + by + c = 0$, use

Using the solution, substitution is done with the centroid $\left(\frac{17}{3}, \frac{28}{3}\right)$ and the reflected image is obtained as

There is a discrepancy in the solution because it mentions the reflection line as $2x + 6y - 53 = 0$ in one place, while the question states $3x + 6y - 53 = 0$. The solution concludes with $h=3$ and $k=4$ and identifies option $D$ as correct.

Now compute

Therefore, $h^2 + k^2 + hk = 37$ and the correct option is D.

Given: The centroid of a triangle is the average of its vertex coordinates.

Find: The value after reflecting that centroid in the given line.

First find the triangle vertices from the pairwise intersections of the three lines. The extracted solution lists them as

So the centroid is the arithmetic mean of the coordinates.

Thus,

A reflection in a line sends a point to its mirror image, and the solution directly evaluates that reflected point as $(3,4)$.

Finally,

So the required value is $37$.