Suppose AB is a focal chord of the parabola y^2=12x of length l and slope m

Question

Suppose $AB$ is a focal chord of the parabola ${y}^{2}=12x$ of length $l$ and slope $m<\sqrt{3}$. If the distance of the chord $AB$ from the origin is $d$, then $l{d}^{2}$ is equal to \_\_\_\_\_\_\_\_\_.

Answer: 108

Accepted Answer

Equation of focal chord

$y-0=	an 	heta .(x-3)$

Distance from origin

$\begin{gathered} d=|\frac{-3	an 	heta }{\sqrt{1+{	an }^{2}	heta }}| \ I=4	imes 3{\csc }^{2}	heta \ I.{d}^{2}=\frac{9{	an }^{2}	heta }{1+{	an }^{2}	heta }	imes 12{\csc }^{2}	heta \ =\frac{108{\csc }^{2}	heta }{1+{\cot }^{2}	heta }=108 \end{gathered}$

Solutions · 1