Showing that level curves of $w=1/z$ are orthogonal at their points of intersection.

Showing that level curves of $w=1/z$ are orthogonal at their points of
intersection.

I have this exercise to show that the level curves of the function
$f(z)=1/z=u(x,y)+iv(x,y)$ are orthogonal to each other at points of
intersection. So I calculated that the level curves $$u(x,y)=1/2a,$$
$$v(x,y)=1/2b,$$
correspond to circles $$C_1:(x-a)^2+y^2=a^2,$$ $$C_2: x^2+(y+b)^2=b^2.$$
There are always two points of intersection (one for either of the
parameters being zero), one at the origin and another one in an
undisclosed location (I tried to find the coordinates but gave up after
producing probably the ugliest quadratic equation I've ever laid my eyes
on). But now, how do I actually show orthogonality? The intersection at
the origin is obvious, but what about the ugly point? This exercise is
really early on too (previous part had me checking C-R conditions), so I
don't think I can just say "well $1/z$ is conformal".