Question

If normals are drawn from the point $P$ whose slopes are $m_1$ and $m_2$. If $m_1\cdot m_2=\alpha$ and point $P$ lies on the parabola $y^2=4x$, then the value of $\alpha$ is

Answer 1

Let any point on the parabola $y^2=4x$ is $P(h,k)$
The equation of normal on the given point is
$\Rightarrow k=mh-2m-m^3$
$\Rightarrow m^3+m(2-h)+k=0$
So,
$m_1{m}_2{m}_3=-k$
$\Rightarrow m_3=-\frac{k}{\alpha }$
Putting $m_3$ in the equation, we get
$\Rightarrow {(-\frac{k}{\alpha })}^3-\frac{k}{\alpha }(2-h)+k=0\Rightarrow k^2={\alpha }^{2}h-2{\alpha }^2+{\alpha }^3$

Thus, the locus of $(h,k)$ is
$y^2={\alpha }^{2}x-2{\alpha }^2+{\alpha }^3$

As $P$ lies on the parabola, so comparing it with $y^2=4x$, we get
${\alpha }^2=4\text{and}-2{\alpha }^2+{\alpha }^3=0\Rightarrow \alpha =2$