Let E denote the parabola y2=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?
(A)The triangle PFQ is a right-angled triangle✓ Correct
(B)The triangle QPQ′ is a right-angled triangle✓ Correct
(C)The distance between P and F is 52
(D)F lies on the line joining Q and Q'✓ Correct
Answer: (A), (B), (D)
Solution
The parabola y2=8x has focus F=(2,0) and directrix x=−2. Point P = (-2, 4) lies on the directrix.
A fundamental property: if tangents from an external point P to a parabola touch at Q and Q', and P lies on the directrix, then the chord QQ' passes through the focus F, and angle QPQ' is a right angle.
(A) Since QQ' passes through F and angle QPQ' = 90°, triangle PFQ is right-angled at P. Correct.
(B) Triangle QPQ' has a right angle at P. Correct.
(C) Distance PF = (−2−2)2+(4−0)2=16+16=42, not 52. Incorrect.
(D) As established, QQ' passes through focus F. Correct.
Correct answers: (A), (B), (D).
JEE Advanced · 2021 · 03 Oct 2021 · Paper 2 · One or more correct options
