JEE Mathematics 2024 Question with Solution

Given: The point is $A = \left(2, 3\right)$ and the line is $2x - 3y + 28 = 0$.

Find: The distance from $A$ to the line, measured parallel to $\sqrt{3}x - y + 1 = 0$.

Take a point $P$ on the required line through $A$ parallel to $\sqrt{3}x - y + 1 = 0$. From the solution, its parametric form is

where $r$ is the required distance.

Since $P$ lies on $2x - 3y + 28 = 0$, substitute its coordinates:

Simplifying,

So,

From the solution, this gives

Therefore, the required distance is $4 + 6\sqrt{3}$ and the correct option is D.

Given: The point is $\left(2, 3\right)$ and the line is $2x - 3y + 28 = 0$.

Find: The distance measured parallel to $\sqrt{3}x - y + 1 = 0$.

The first approach in the solution contains inconsistent intermediate working, but it explicitly concludes that the correct answer is $4 + 6\sqrt{3}$. The second approach also states The Correct Answer is $4 + 6\sqrt{3}$.

Since both solution blocks identify the same final value, we take the required answer as $4 + 6\sqrt{3}$, corresponding to option D.