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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?

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PQ₁ = 3

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

SQ₂ = 1

SQ₁ = 2

answer is B, C, D.

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Detailed Solution

Slopes of tangents from P(-2, 1) are -1 and 1/2
points of Contact are p₁(4, 4) and p₂(1, -2)
Equation of sp₁ is 4x - 3y - 4 = 0
$Q_{1} = (\frac{2}{5}, \frac{- 4}{5})$
Equation of sp₂ is x - 1 = 0
Q₂ = (1, 1)
Now check options

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