JEE Advanced 2022 · Paper 1 · Mathematics Q13

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MathematicsMultiple correct+4 / -2Question 13 of 54

Consider the parabola $y^2 = 4x$ . 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 $SP_1$ and $SP_2$ respectively such that $PQ_1$ is perpendicular to $SP_1$ and $PQ_2$ is perpendicular to $SP_2$ . Then, which of the following is/are TRUE?

SQ_1 = 2

Q_1Q_2 = \dfrac{3\sqrt{10}}{5}

PQ_1 = 3

SQ_2 = 1

Answer and worked solution

Official answer: B, C, D

Points of contact. The focus is $S=(1,0)$ . A tangent $y=mx+\frac1m$ through $P(-2,1)$ needs $2m^2+m-1=0$ , so $m=\frac12$ or $-1$ . The points of contact $\left(\frac1{m^2},\frac2m\right)$ are $(4,4)$ and $(1,-2)$ .

Line through (4, 4). $S$ to $(4,4)$ has unit direction $(\frac35,\frac45)$ , and $P-S=(-3,1)$ .

Projection: $SQ=|(-3)\tfrac35+1\cdot\tfrac45|=1$ .
Perpendicular: $PQ=3$ .
Foot: $Q=(\frac25,-\frac45)$ .

Line through (1, −2). This is $x=1$ , so the foot is $Q=(1,1)$ , with $SQ=1$ and $PQ=3$ .

Options. Whichever point is called $P_1$ :

$SQ_1=SQ_2=1$ , so (A) is false and (D) is true.
$PQ_1=3$ , so (C) is true.
$Q_1Q_2=\sqrt{(\frac35)^2+(\frac95)^2}=\frac{3\sqrt{10}}5$ , so (B) is true.

Topics:Tangents and normals to conics Parabola Distance of a point from a line

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JEE Advanced 2022 · Paper 1, Mathematics question 13