JEE Advanced · Mathematics · 13. Parabola

Consider the parabola $y^{2} = 4 x$ . Let $S$ be the focus of the parabola. A pair of tangents drawn to the parabola from the point $P = (- 2, 1)$ meet the parabola at $P_{1}$ and $P_{2}$ . Let $Q_{1}$ and $Q_{2}$ be points on the lines $S P_{1}$ and $S P_{2}$ respectively such that $P Q_{1}$ is perpendicular to $S P_{1}$ and $P Q_{2}$ is perpendicular to $S P_{2}$ . Then, which of the following is/are TRUE?

A $S Q_{1} = 2$
B $Q_{1} Q_{2} = \frac{3 \sqrt{10}}{5}$
C $P Q_{1} = 3$
D $S Q_{2} = 1$

Verified Solution

Answer & Solution

Correct Answer

(B) $Q_{1} Q_{2} = \frac{3 \sqrt{10}}{5}$

Step-by-step Solution

Detailed explanation

On plotting the diagram of the given value we have, Now let

P_{1} (t^{2}, 2 t)

then tangent at

P_{1}

will be

t y = x + t^{2}

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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