Equation of a common tangent to the parabola

Let the equation of tangent to parabola

$y^{2}=4 x$ be $y=m x+\frac{1}{m}$

It is also a tangent to hyperbola $x y=2$

$\Rightarrow x\left(m x+\frac{1}{m}\right)=2$

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

$\mathrm{D}=0 \Rightarrow \mathrm{m}=-\frac{1}{2}$

So tangent is $2 y+x+4=0$