Question

The circles $x^2+y^2-2\lambda x+c^2=0$ and $x^2+y^2+2\lambda y=0$ will have three common tangents if
$|c:\lambda |$ is

• A. $\sqrt{\sqrt{3}-1}:1$
• B. $\sqrt{2\sqrt{2}-2}:1$
• C. $3:2$
• D. none of these

Answer 1

Answer: (B)

$\sqrt{2\sqrt{2}-2}:1$

$C_1=(\lambda ,0),r_1=\sqrt{{\lambda }^2-c^2},C_2=(0,-\lambda ),r_2=\sqrt{{\lambda }^2}=|\lambda |$
For three common tangent $C_1{C}_2=r_1+r_2$
$\Rightarrow \sqrt{{\lambda }^2+{\lambda }^2}=|\lambda |+\sqrt{{\lambda }^2-c^2}$
$\Rightarrow |\lambda |(\sqrt{2}-1)=\sqrt{{\lambda }^2-c^2}$
$\Rightarrow \sqrt{1-(c/\lambda {)}^2}=(\sqrt{2}-1)$
$\Rightarrow |c:\lambda |=\sqrt{2\sqrt{2}-2}:1$