Question

The number of common tangents to circle $x^{2}+y^{2}+2 x+8 y-23=0$ and $x^{2}+y^{2}-4 x-10 y+9=0$, is.

• A. 1
• B. 3
• C. 2
• D. none of these

Answer 1

Answer: (C)

2

Answer 2

$S 1: x ^{2}+ y ^{2}+2 x +8 y -23=0$ $C_{1}=(-1,-4), r_{1}^{2}=\sqrt{(-1)^{2}+(-4)^{2}+23}=2 \sqrt{10}$ S2 : $x^{2}+y^{2}-4 x-10 y+9=0$ $C_{2}=(2,5), r_{2}^{2}=\sqrt{(2)^{2}+(5)^{2}-9}=2 \sqrt{5}$ $C_{1} C_{2}=\sqrt{3^{2}+9^{2}}=\sqrt{90}=3 \sqrt{10}$ $r_{1}+r_{2}=2 \sqrt{10}+2 \sqrt{5}=\sqrt{10}(2+\sqrt{2})$ $3 \sqrt{10}(2-\sqrt{2}) \sqrt{10}$ $\left| r _{1}- r _{2}\right|< C _{1} C _{2}< r _{1}+ r _{2}$ Only $2$ common tangents.