Area of an equilateral triangle inscribed in the parabola y^2 = 8x  with one of its vertices on the vertex of this parabola is: 
Option: 1 128\sqrt3
Option: 2 192\sqrt3
Option: 3 64\sqrt3
Option: 4 256\sqrt3

Answers (1)

Let the two vertices of the triangle be Q and R. Points Q and R will have the same x - coordinate =k( say )

Now in the right triangle PRT, right-angled at T.

\tan 30^{\circ}=\frac{\mathrm{RT}}{\mathrm{k}} \Rightarrow \frac{1}{\sqrt{3}}=\frac{\mathrm{RT}}{\mathrm{k}} \Rightarrow \mathrm{RT}=\frac{\mathrm{k}}{\sqrt{3}} \Rightarrow \mathrm{R}\left(\mathrm{k}, \frac{\mathrm{k}}{\sqrt{3}}\right)

Now R lies on the parabola: y^{2}=4 ax

\\\Rightarrow\left(\frac{k}{\sqrt{3}}\right)^{2}=4 a(k) \\ \\\Rightarrow \frac{k}{3}=4 a \\ \\\Rightarrow k=12 a

\\\text { Length of side of the triangle }=2(\mathrm{RT})=2 \cdot \frac{\mathrm{k}}{\sqrt{3}}=2 \cdot \frac{(12 \mathrm{a})}{\sqrt{3}}=8 \sqrt{3} \mathrm{a}=16\sqrt{3}

\text { Area of Equilateral triangle is }= \frac{\sqrt{3}}4a^2 =192\sqrt{3}

 

Most Viewed Questions

Preparation Products

Knockout JEE Main April 2021 (One Month)

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Weekend Live Classes, Mentorship from our Experts, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 14000/- ₹ 4999/-
Buy Now
Knockout JEE Main May 2021

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Weekend Live Classes, Mentorship from our Experts, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 22999/- ₹ 9999/-
Buy Now
Test Series JEE Main May 2021

Unlimited Chapter Wise Tests, Unlimited Subject Wise Tests, Unlimited Full Mock Tests, Get Personalized Performance Analysis Report,.

₹ 6999/- ₹ 2999/-
Buy Now
Knockout JEE Main May 2022

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Weekend Live Classes, Mentorship from our Experts, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 34999/- ₹ 14999/-
Buy Now
JEE Main Rank Booster 2021

Booster and Kadha Video Lectures, Unlimited Full Mock Test, Adaptive Time Table, 24x7 Doubt Chat Support,.

₹ 13999/- ₹ 6999/-
Buy Now
Boost your Preparation for JEE Main 2021 with Personlized Coaching
 
Exams
Articles
Questions