JEE Advanced 2014 · Paper 1, Mathematics question 49

Write $S:\ x^2+y^2+2gx+2fy+c=0$. Orthogonality of two circles means $2g_1g_2+2f_1f_2=c_1+c_2$.

• With $x^2+y^2-1=0$: $0=c-1$, so $c=1$.
• With $x^2+y^2-2x-15=0$: $2g(-1)=c-15=-14$, so $g=7$.
• Through $(0,1)$: $1+2f+1=0$, so $f=-1$.

Centre and radius. Centre $(-g,-f)=(-7,1)$; radius $\sqrt{49+1-1}=7$.

(B), (C)