Let, point of intersection of the tangents is $\left(h,k\right).$

Chord of contact through $\left(h,k\right)$ for the circle $x^2+y^2=r^2$ is

$hx+ky-r^2=0$ which touches the circle $x^2+y^2+2rx=0$

$\Rightarrow$ The distance of $hx+ky-r^2=0$ from the centre $\left(-r,0\right)$ of circle $x^2+y^2+2rx=0$

is equal to the radius $\left(r\right)$ of the circle $x^2+y^2+2rx=0$

$\Rightarrow \frac{\left|h\left(-r\right)+k\left(0\right)-r^2\right|}{\sqrt{h^2+k^2}}=\left|r\right|$

$\Rightarrow \left|h+r\right|=\sqrt{h^2+k^2}$

$\Rightarrow x^2+2rx+r^2=x^2+y^2\Rightarrow y^2=2r\left(x+\frac{r}{2}\right)$ is the required locus