The equation of a common tangent to the curves,$y^2=16x$ and $xy=-4$, is

Equation of tangent to the parabola

$y^2=16x$ is

$y=mx+\frac{4}{m}$                                                                          …(i)

it is also the tangent to the curve $xy=-4$                           …(ii)

Solving (i) and (ii), we get

$\frac{-4}{x}=mx+\frac{4}{m}\Rightarrow m{x}^2+\frac{4}{m}x+4=0$

For tangents the roots are equal.

$\therefore \frac{16}{m^2}-4.4m=0\Rightarrow m^3=1\Rightarrow m=1$

So, equation of common tangent is $y=x+4$