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

A tangent to the parabola $y^2=16x$ can be written as:
$y=mx+\frac{4}{m}$ …. (1)
It will touch the hyperbola $xy= –4,$ if the quadratic $x\left(mx+\frac{4}{m}\right)=–4$ has coincident roots,
$\Rightarrow \left(m^2\right)x^2+4x+4m=0, D\equiv 16-16m^3=0$
$\Rightarrow m=1$
$\Rightarrow$ Equating of common tangent will be $y=x+4$
$\Rightarrow x-y+4=0$