Find the equation of the common tangent to the parabola $y^2=16x$ and the circle $x^2+y^2=8$.

Let the equation of the tangent to the parabola

$y^2=16x$

$y=mx+\frac{4}{m}$

$\Rightarrow m^{2}x-my+4=0$    …(i)

If the Eq. (i) be a tangent to the circle $x^2+y^2=8$, the

length of the perpendicular from the center to the tangent is equal to the radius of the circle. 

Therefore, 

$\begin{array}{l}\left|\frac{0-0+4}{\sqrt{m^4+m^2}}\right|=2\sqrt{2}\\ \Rightarrow 8\left(m^4+m^2\right)=16\\ \Rightarrow \left(m^4+m^2-2\right)=0\\ \Rightarrow \left(m^4+2m^2-m^2-2\right)=0\\ \Rightarrow \left(m^2+2\right)\left(m^2-1\right)=0\\ \Rightarrow \left(m^2-1\right)=0\\ \Rightarrow m=\pm 1\end{array}$

Hence, the common tangents are $y=x\pm 4$.