$\newcommand\E{\mathscr E}$The expectation in question is $I/c_1$, where $$I:=E\frac{a^2}{a^2+cb^2},\quad c:=c_2/c_1>0.$$ In turn, using polar coordinates, we get $$I=\frac1{2\pi}\int_0^{2\pi}\frac{\cos^2 t\, dt}{\cos^2 t+c\sin^2 t},\quad c:=c_2/c_1>0.$$ Further, writing $$I=\frac4{2\pi}J$$ for $$J:=\int_0^{\pi/2}\frac{\cos^2 t\, dt}{\cos^2 t+c\sin^2 t}$$ and using the standard substitution $t=\arctan u$, so that $\cos^2 t=1/(1+u^2)$ and $\sin^2 t=u^2/(1+u^2)$, we have $$J=\int_0^\infty\frac{du}{(1+cu^2)(1+u^2)}.$$ Using partial fraction decomposition to compute $J$ and collecting the pieces, we finally get that the expectation in question is $$\E(c_1,c_2):=\frac1{c_1+\sqrt{c_1c_2}}.$$
"Sanity" checks: (i) $\E(tc_1,tc_2)=\E(c_1,c_2)/t$ for all real $t>0$; (ii) $\E(1,0)=1$; (iii) $\E(1,1)=1/2$; (iv) $\E(c_1,c_2)\to\infty$ as $c_1\downarrow0$.