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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 $$\frac1{c_1+\sqrt{c_1c_2}}.$$