Question

What is the length of focal chord of a parabola $y^2=8xmakingangle{30}^{\circ }$ with x axis?

Answer 1

The parabola described in the question can be represented as in the figure.

Let the ends of focal chord be.

$A\equiv (a{t}^2,2at)$

$B\equiv [\frac{a}{t^2},-\frac{2a}{t}]$

Slope of $AB=\frac{2at+\frac{2a}{t}}{a{t}^2-\frac{a}{t^2}}$

$tan\alpha =\frac{2.(t+\frac{1}{t})}{t^2-\frac{1}{t^2}}=\frac{2}{t-\frac{1}{t}}=\frac{2t}{t^2-1}$....................................(i)

we know that,

$tan\alpha =\frac{2tan\left(\frac{\alpha }{2}\right)}{1-ta{n}^2\left(\frac{\alpha }{2}\right)}$

if we assign $tan=\frac{\alpha }{2}=\frac{1}{t}$, then expression (1) is satisfied.

$\therefore t=cot\frac{\alpha }{2}$

Length of $AB=a(t+\frac{1}{t})\sqrt{4+{(t-\frac{1}{t})}^2}$

$=a(t+\frac{1}{t})\sqrt{{(t+\frac{1}{t})}^2}=a{(t+\frac{1}{t})}^2$

$=a.[cot\frac{\alpha }{2}+tan\frac{\alpha }{2}{]}^2=4a.cose{c}^2\alpha$

$=4\times 2\times 4=32$