Mathematics · Parabola

JEE Advanced 2024 — Paper 2 — Question 7

Let $A_{1}, B_{1}, C_{1}$ be three points in the $x y$ -plane. Suppose that the lines $A_{1} C_{1}$ and $B_{1} C_{1}$ are tangents to the curve $y^{2}=8 x$ at $A_{1}$ and $B_{1}$ , respectively. If $O=(0,0)$ and $C_{1}=(-4,0)$ , then which of the following statements is(are) TRUE?

Option A:
The length of the line segment $\mathrm{OA}_{1}$ is $4 \sqrt{3}$
Correct
Option B:
The length of the line segment $A_{1} B_{1}$ is 16
Option C:
The orthocenter of the triangle $\mathrm{A}_{1} \mathrm{~B}_{1} \mathrm{C}_{1}$ is $(0,0)$
Correct
Option D:
The orthocenter of the triangle $A_{1} B_{1} C_{1}$ is $(1,0)$

Answer: A, C

Step-by-step solution

Let $\mathrm{OA}_{1}$ be $\mathrm{y}=\mathrm{mx}+\mathrm{c}$

$\Rightarrow 0=-4 \mathrm{~m}+\mathrm{c}$ or $\mathrm{c}=4 \mathrm{~m}$

Also, $2=\mathrm{mc}$

$\therefore \mathrm{m}= \pm \frac{1}{\sqrt{2}}, \mathrm{c}= \pm 2 \sqrt{2}$

$\frac{1}{\mathrm{t}}=\mathrm{m}$

$\Rightarrow \mathrm{A}_{1}(4,4 \sqrt{2})$ and $\mathrm{B}_{1}(4,-4 \sqrt{2})$

Orthocentre is $(0,0)$

$\mathrm{A}_{1} \mathrm{~B}_{1}=8 \sqrt{2}$

$\mathrm{OA}_{1}=4 \sqrt{3}$

Answer key and solution verified before publishing.

Practise Parabola

Start with this question, then two more from the same chapter — with a tutor that explains every step. Free.

Solve a similar one free

Exam: JEE Advanced 2024
Paper: Paper 2
Subject: Mathematics
Chapter: Parabola
Topic: Pole and Polar wrt a Parabola

A normal with slope frac1√(6) is drawn from the point (0,-alpha) to the parabola x^2=-4 ay, where a 0 . Let L be the line passing through…

Answer: A, C

Step-by-step solution

Practise Parabola

More Parabola questions from this paper