Consider the parabola . Let be the focus of the parabola. A pair of tangents drawn to the parabola from the point meet the parabola at and Let and be points on the lines and respectively such that and Then, which of the following is/are TRUE?

Question

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 $P Q_{1} ⊥ S P_{1}$ and $P Q_{2} ⊥ S P_{2} .$ Then, which of the following is/are TRUE? [2022]

Smart Achievers · Accepted Answer

(1, 2, 3, 4)

Let a rough graph to refer the question

Let parametric point at $P_{1} (t^{2}, 2 t)$ , then tangent at $P_{1}$

$t y = x + t^{2}$

Since it passes through $(- 2, 1)$

$S P_{2}$ $t^{2} - t - 2 = 0$

$∴$ $t = 2, - 1$

$∴$ $P_{1} (4, 4) and P_{2} (1, - 2)$

$∴$ $S P_{1} : 4 x - 3 y - 4 = 0 and S P_{2} : x - 1 = 0$

and for $Q_{1} :$ $\frac{x_{1} + 2}{4} = \frac{y_{1} - 1}{- 3} = \frac{- (- 8 - 3 - 4)}{25} = \frac{3}{5}$

$∴$ $x_{1} = \frac{2}{5}, y_{1} = - \frac{4}{5} and Q_{2} = (1, 1)$

$So, S Q_{1} = \sqrt{{(1 - \frac{2}{5})}^{2} + {(\frac{4}{5})}^{2}} = 1$

$Q_{1} Q_{2} = \sqrt{\frac{9}{25} + \frac{81}{25}} = \sqrt{\frac{90}{25}} = \frac{3 \sqrt{10}}{5}$

$P Q_{1} = \sqrt{\frac{144}{25} + \frac{81}{25}} = \frac{15}{5} = 3, S Q_{2} = 1$

Solution

1 $S Q_{1} = 2$

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

3 $P Q_{1} = 3$

4 $S Q_{2} = 1$

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Solution

1 SQ1=2

2 Q1Q2=3105

3 PQ1=3

4 SQ2=1

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1 $S Q_{1} = 2$

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

3 $P Q_{1} = 3$

4 $S Q_{2} = 1$