Question 1631
CBSE Class 11 Mathematics · Conic Sections · Hyperbolas
An equilateral triangle is inscribed in y²=4ax with one vertex at the parabola's vertex. Find its side length.
Solution to Question 1631
Complete workingTranslate every condition into an equation and retain exact values throughout the conic calculation.
\[\begin{aligned}B &= (x,y),\ C\\&= (x,-y),\quad BC\\&= 2|y|\end{aligned}\]
\[\begin{aligned}OB &= BC\Rightarrow x^2+y^2\\&= 4y^2\Rightarrow x^2\\&= 3y^2\end{aligned}\]
\[\begin{aligned}y^2 &= 4ax\Rightarrow x^2\\&= 12ax\Rightarrow x\\&= 12a\end{aligned}\]
\[\begin{aligned}y^2 &= 48a^2\Rightarrow BC\\&= 2|y|\\&= 8\sqrt3a\end{aligned}\]
The last line satisfies the original conditions and supplies every requested part.
Final answer
\[8\sqrt3a\]