= Solution
Write the constant <spherical harmonic> $Y_{00}=C$. Its <orthonormality> gives $4\pi|C|^2=1$, independently of its phase convention. <Orthogonality> to $Y_{00}$ gives
$$
\int_{S^2}\overline{Y_{\ell m}(\Omega)}\,d\Omega=0\quad(\ell>0),\qquad
\int_{S^2}\overline{Y_{00}(\Omega)}Y_{00}(\Omega_0)\,d\Omega=1.
$$
Only the $\ell=m=0$ term contributes to the <surface integral> of the <surface charge density>. Multiplying by the surface-area factor $a^2$ gives
$$
Q_a=-qa^2\frac{a^{-1}}{1-a/b}\left(\frac1{r_0}-\frac1b\right).
$$
Thus the <spherical harmonic> calculation reproduces the total induced <electric charge>:
$$
\boxed{Q_a=-q\frac{a(b-r_0)}{r_0(b-a)}.}
$$
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