JEE Advanced · Mathematics · 13. Parabola

Let $E$ denote the parabola $y^{2} = 8 x .$ Let $P = (- 2, 4)$ , and let $Q$ and $Q^{'}$ be two distinct points on $E$ such that the lines $P Q$ and $P Q^{'}$ are tangents to $E$ . Let $F$ be the focus of $E$ . Then which of the following statements is (are) TRUE?

A The triangle $P F Q$ is a right-angled triangle
B The triangle $Q P Q^{'}$ is a right-angled triangle
C The distance between $P$ and $F$ is $5 \sqrt{2}$
D $F$ lies on the line joining $Q$ and $Q^{'}$

Verified Solution

Answer & Solution

Correct Answer

(A) The triangle $P F Q$ is a right-angled triangle

Step-by-step Solution

Detailed explanation

y^{2} = 8 x

\Rightarrow a = 2,

Focus

F (2, 0),

and Directrix

x = - 2

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.

Column 1	Column 2	Column 3
(I) $x^{2} + y^{2} = a^{2}$	(i) $m y = m^{2} x + a$	(P) $(\frac{a}{m^{2}}, \frac{2 a}{m})$
(II) $x^{2} + a^{2} y^{2} = a^{2}$	(ii) $y = m x + a \sqrt{m^{2} + 1}$	(Q) $(\frac{- m a}{\sqrt{m^{2} + 1}}, \frac{a}{\sqrt{m^{2} + 1}})$
(III) $y^{2} = 4 a x$	(iii) $y = m x + \sqrt{a^{2} m^{2} - 1}$	(R) $(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} + 1}}, \frac{1}{\sqrt{a^{2} m^{2} + 1}})$
(IV) $x^{2} - a^{2} y^{2} = a^{2}$	(iv) $y = m x + \sqrt{a^{2} m^{2} + 1}$	(S) $(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} - 1}}, \frac{- 1}{\sqrt{a^{2} m^{2} - 1}})$

If a tangent to a suitable conic (Column 1) is found to be

y = x + 8

and its point of contact is

(8, 16)

, then which of the following options is the only Correct combination?

JEE Advanced 2017 Hard

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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?

Step-by-step Solution

Let $E$ denote the parabola $y^{2} = 8 x .$ Let $P = (- 2, 4)$ , and let $Q$ and $Q^{'}$ be two distinct points on $E$ such that the lines $P Q$ and $P Q^{'}$ are tangents to $E$ . Let $F$ be the focus of $E$ . Then which of the following statements is (are) TRUE?