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Q.
Length of the focal chord of the parabola $y^2 = 4ax$ at a distance p from the vertex is :
Conic Sections
Solution:
Length = $\sqrt{\left(2 at + \frac{2a}{1}\right)^{2}+\left(at^{2}-\frac{a}{t^{2}}\right)^{2}}=\frac{a\left(1+t^{2}\right)^{2}}{t^{2}}$
Now equation of focal chord, $2 tx + y (1 - t^2) - 2$ at $= 0$
$\Rightarrow\left|\frac{2at}{1+t^{2}}\right|\Rightarrow\frac{4a^{2}}{P^{2}}=\frac{\left(1+t^{2}\right)^{2}}{t^{2}}$
