JEE Mathematics 2025 Question with Solution

Given: The circle $x^2 + y^2 - 2x + 4y - 4 = 0$ is reflected in the line $2x - 3y + 5 = 0$. Point $A$ lies on the reflected circle $C$ such that $OA$ is parallel to the x-axis and $A$ is on the right-hand side of the centre $O$ of $C$.

Find: The required value asked in the problem, as concluded from the provided solution.

First rewrite the given circle in standard form by completing the square:

So the original circle has centre $(1,-2)$ and radius $3$.

From the solution, the centre is then reflected across the line $2x - 3y + 5 = 0$, and the reflected circle is stated as

Hence the reflected circle has centre $O = (-3,4)$ and radius $3$.

Since $OA$ is parallel to the x-axis and $A$ lies on the right-hand side of the centre, $A$ is the rightmost point of the reflected circle. Therefore its x-coordinate is centre x-coordinate plus radius:

However, the provided the solution explicitly concludes with The Correct Option is D and the final answer $4$.

So, the correct option is D.

Therefore, the required value is $4$.

Note: The working shown in the solution is internally inconsistent with the stated geometry, but the solution explicitly marks option D as correct, so the extracted answer is D.

Given:

• Original circle: $x^2 + y^2 - 2x + 4y - 4 = 0$
• Reflection line: $2x - 3y + 5 = 0$

Find: The final value indicated by the provided solution.

The solution rewrites the circle as

so the original centre is $(1,-2)$ and radius is $3$.

In the second approach, the solution states that after reflection the new circle is

which means the reflected centre is $(-3,4)$ and the radius remains $3$.

The same the solution then gives the final statement: The correct option is $\boxed{4}$.

Therefore, the extracted answer is $4$, i.e. option D.