JEE Advanced 2021 Paper 2Question 1
Let $E$ denote the parabola $y^2=8x$. Let $P=(-2,4)$, and let $Q$ and $Q'$ be two distinct points on $E$ such that the lines $PQ$ and $PQ'$ are tangents to $E$. Let $F$ be the focus of $E$. Then which of the following statements is (are) TRUE ?
Parabola
Solution
Concept. $P$ lies on the directrix, so the two tangents from $P$ are perpendicular and the chord of contact passes through the focus. Formulas used. (1) $y^2=4ax$: focus $(a,0)$, directrix $x=-a$.(2) Chord of contact from $(x_1,y_1)$: $yy_1=2a(x+x_1)$.(3) Tangent $y=mx+\frac am$.Step 1 - Basics. $y^2=8x$: $a=2$, $F=(2,0)$, directrix $x=-2$. So $P(-2,4)$ is on the directrix. Step 2 - Statement (B). By (3) through $P$: $4=-2m+\frac2m\Rightarrow m^2+2m-1=0$. Product of slopes $=-1$, so $PQ\perp PQ'$. (B) is true. Step 3 - Statement (D). By (2): $4y=4(x-2)\Rightarrow y=x-2$. At $F(2,0)$: $0=2-2$. So $F$ lies on $QQ'$. (D) is true.Step 4 - Statement (A). Slope of $PF=\frac{4-0}{-2-2}=-1$, slope of $QQ'=1$. Product $-1$, so $PF\perp FQ$ and $\angle PFQ=90^\circ$. (A) is true. Step 5 - Statement (C). $PF=\sqrt{4^2+4^2}=4\sqrt2\ne5\sqrt2$. (C) is false. Answer: A, B, D