JEE Advanced 2021 · Paper 2 · Mathematics Q6

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MathematicsMultiple correct+4 / -2Question 44 of 57

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?

The triangle

PFQ

is a right-angled triangle

The triangle

QPQ'

is a right-angled triangle

The distance between

P

and

F

5\sqrt2

F

lies on the line joining

Q

and

Q'

Answer and worked solution

Official answer: A, B, D

Setup. For $y^2=8x$ , $a=2$ : the focus is $F=(2,0)$ and the directrix is $x=-2$ . So $P=(-2,4)$ lies on the directrix.

Facts for a point on the directrix.

The two tangents from it are perpendicular.
Its chord of contact, $4y=4(x-2)$ , passes through the focus.
$FP\perp QQ'$ .

Check each option.

(A) Since $FP\perp QQ'$ and $F$ lies on $QQ'$ , $\angle PFQ=90^\circ$ . True.
(B) The tangents are perpendicular, so $\angle QPQ'=90^\circ$ . True.
(C) $PF=\sqrt{16+16}=4\sqrt2$ , not $5\sqrt2$ . False.
(D) $F$ lies on the chord of contact $QQ'$ . True.

(A), (B), (D)

Topics:Parabola Tangents and normals to conics

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