Number of normals drawn through the point (8, 4) to the parabola ${\mathrm{y}}^2=2\mathrm{x}$

$$\begin{gathered} {\mathrm{y}}^2=2\mathrm{x} \\ \Rightarrow 4\mathrm{a}=2 \\ \Rightarrow a=\frac{1}{2} \end{gathered}$$

Given point (8, 4)  

Equation of normal to   ${\mathrm{y}}^2=4\mathrm{ax} is \mathrm{y}=\mathrm{mx}-2\mathrm{am}-{\mathrm{am}}^3$ 

it is passing through (8, 4)

$$\begin{gathered} 4=8\mathrm{m}-\mathrm{m}-\frac{1}{2}{\mathrm{m}}^3 \\ 4=7\mathrm{m}-\frac{{\mathrm{m}}^3}{2} \\ 8=14\mathrm{m}-{\mathrm{m}}^3 \\ {\mathrm{m}}^3-14\mathrm{m}+8=0 \end{gathered}$$

This Equation has 3 distinct Real Roots 

So,number of normals = 3