The length of the focal chord of the parabola | vedclass

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(C) Let the focal chord make an angle $	heta$ with the $x$-axis. The length of the focal chord is given by $L = 4a \csc^2 	heta$.The distance $p$ of

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$\frac{4a^3}{p^2}$

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(C) Let the focal chord make an angle $	heta$ with the $x$-axis. The length of the focal chord is given by $L = 4a \csc^2 	heta$.The distance $p$ of the focal chord from the vertex $(0,0)$ is the perpendicular distance from the origin to the line passing through the focus $(a,0)$ with inclination $	heta$.The equation of the focal chord is $(x - a) \sin 	heta - y \cos 	heta = 0$.The distance $p$ from the origin $(0,0)$ is $p = |\frac{-a \sin 	heta}{\sqrt{\sin^2 	heta + \cos^2 	heta}}| = |a \sin 	heta|$.Thus,$\sin 	heta = \frac{p}{a}$,which implies $\csc 	heta = \frac{a}{p}$.Substituting this into the length formula: $L = 4a (\frac{a}{p})^2 = \frac{4a^3}{p^2}$.

The length of the focal chord of the parabola $y^2 = 4ax$ at a distance $p$ from the vertex is:

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