Question

The two circles $x^2+y^2-2 x+6 y+6=0$ and $x^2+y^2-5 x+6 y+15=0$

(a) intersect

(b) concentric

(c) touch internally

(d) touch externally

Answer 1

SOLUTION — Here,

$\begin{aligned}C_1(1,-3), r_1 & =\sqrt{1+9-6}=2 \\C_2\left(\frac{5}{2},-3\right), r_2 & =\sqrt{\frac{25}{4}+9-15}=\frac{1}{2} \\C_1 C_2 & =\frac{3}{2} \\r_1-r_2 & =\frac{3}{2} \\C_1 C_2 & =r_1-r_2\end{aligned}$

$\begin{array}{l}r_1-r_2=\frac{3}{2} \\C_1 C_2=r_1-r_2\end{array}$

Hence, both circles touch internally

So, The correct option of this question will be (C).