Question

The angle between the circles $S_1:x^2+y^2-4x+6y+11=0$ and $S_2:x^2+y^2-2x+8y+13=0$ is

• A. ${90}^{\text{o}}$
• B. ${60}^{\text{o}}$
• C. ${45}^{\text{o}}$
• D. ${30}^{\text{o}}$

Answer 1

The correct option is C ${45}^{\text{o}}$

$S_1:x^2+y^2-4x+6y+11=0$
$\Rightarrow C_1(2,-3),r_1=\sqrt{2}$
$S_2:x^2+y^2-2x+8y+13=0$
$\Rightarrow C_2(1,-4),r_2=2$

Distance between centres
$d=|C_1{C}_2|$
$d=\sqrt{(2-1{)}^2+(-3+4{)}^2}=\sqrt{2}$

If angle between the circles is $\theta$, then
$\cos \theta =\left(\frac{r_1^2+r_2^2-d^2}{2r_1{r}_2}\right)$
$\Rightarrow \cos \theta =\left(\frac{2+4-2}{2\times \sqrt{2}\times 2}\right)=\frac{1}{\sqrt{2}}$
$\Rightarrow \theta ={45}^{\circ }$.