value of $a$ in Normal equation

Question

From a point $(\cos \alpha,\sin \alpha)$, If three normal can be drawn to the prabola $y^2=4ax$, then the value of $a$ is

Try: Equation of Normal at point $P(at^2,2at)$ is $y=-tx+2at+at^3$.

It also passes through the point $(\cos \alpha,\sin \alpha)$

So $\sin \alpha=-(\cos \alpha)t+2at+at^3$.

now I did not understand how can I find value of $a$ from that equation. Thanks!

Answer 1

Hint: You have the equation of normal as $f(t)=at^3+t(2a-\cos\alpha)-\sin\alpha=0$.

Now, for a cubic polynomial to have three distinct real roots,

1. $f'(t)=0$ should have two distinct real roots (say $t_1$ and $t_2$).

2. $f(t_1)\cdot f(t_2)<0$.