The number of common tangents to the circles x^2+y^2-4x-6y-12=0 and x^2+y^2+6x+18y+26=0 is…

Question

The number of common tangents to the circles $x^2+y^2-4x-6y-12=0$ and $x^2+y^2+6x+18y+26=0$ is:

Accepted Answer

Circle 1: centre $(2,3)$, $r_1=\sqrt{4+9+12}=5$.
Circle 2: centre $(-3,-9)$, $r_2=\sqrt{9+81-26}=\sqrt{64}=8$.

Distance between centres $=\sqrt{(2+3)^2+(3+9)^2}=\sqrt{25+144}=13=r_1+r_2$.

Since $d=r_1+r_2$, the circles touch each other externally, giving exactly 3 common tangents.

Solutions · 1