If the pole of a line w. r. t the circle $x^2+y^2=a^2$ lies on the circle $x^2+y^2=a^4$ then the line touches the circle.

A point on $x^2+y^2=a^4$ is $P(a^2\cos \theta ,a^2\sin \theta )$

Polar of $P(a^2\cos \theta ,a^2\sin \theta )$ with respect to $x^2+y^2=a^2$ is $x{a}^2\cos \theta +y{a}^2\sin \theta =a^2$

$\Rightarrow x\cos \theta +y\sin \theta =1$

This line touches $x^2+y^2=1(\because r=d)$