Q.Find k if the pairs of circles $x^2+y^2+4x+8=0$, $x^2+y^2-16y+k=0$ are orthogonal.

Two circles $x^2+y^2+2g_1x+2f_1y+c_1=0$ and $x^2+y^2+2g_2x+2f_2y+c_2=0$ are orthogonal when $2g_1g_2+2f_1f_2=c_1+c_2$.

First circle $x^2+y^2+4x+8=0$: $g_1=2$, $f_1=0$, $c_1=8$.

Second circle $x^2+y^2-16y+k=0$: $g_2=0$, $f_2=-8$, $c_2=k$.

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