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The circle \( S_{1} \) with centre \( C_{1}\left(a_{1}, b_{1}\right) \) and radius \( r_{1} \) touches externally the circle \( S_{2} \) with centre \( C_{2}\left(a_{2}, b_{2}\right) \) and radius \( r_{2} \). If the tangent at their common point passes through the origin, then
(A) \( \left(a_{1}^{2}+a_{2}^{2}\right)+\left(b_{1}^{2}+b_{2}^{2}\right)=r_{1}^{2}+r_{2}^{2} \)
(B) \( \left(a_{1}^{2}-a_{2}^{2}\right)+\left(b_{1}^{2}-b_{2}^{2}\right)=r_{1}^{2}-r_{2}^{2} \)
(C) \( \left(a_{1}^{2}-b_{2}^{2}\right)+\left(a_{2}^{2}+b_{2}^{2}\right)=r_{1}^{2}+r_{2}^{2} \)
(D) \( \left(a_{1}^{2}-b_{1}^{2}\right)+\left(a_{2}^{2}+b_{2}^{2}\right)=r_{1}^{2}+r_{2}^{2} \)
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