Question

Length of focal chord of a parabola y^2 =4ax making an angle theta with the axis is

Answer 1

we have to find length of focal chord of a parabola y² = 4ax making an angle $\theta$ with the axis.

Let two points (at², 2at) and (a/t², 2a/t) in parabola in such a way that line joining the points is passing through focus. the this line is not other than focal chord.
so, length of focal chord , L =
$a\sqrt{\left(t^2-\frac{1}{t^2}\right)^2+4\left(t-\frac{1}{t}\right)^2}\\\\=a\left(t+\frac{1}{t}\right)^2$

now slope of line = $tan\theta=\frac{2t-\frac{2}{t}}{t^2-\frac{1}{t^2}}$

$tan\theta=\frac{2}{t+\frac{1}{t}}$

$t+\frac{1}{t}=2cot\theta$

now,
$\left(t-\frac{1}{t}\right)^2=\left(t+\frac{1}{t}\right)^2+4\\\\=4cot^2\theta+4=4cosec^2\theta$

so, length of focal chord =$4acosec^2\theta$