The length of the tangent drawn from any point on the circle $x^{2}+y^{2}+2 f y+\lambda=0$ to the circle $\mathrm{x}^{2}+\mathrm{y}^{2}+2 \mathrm{fy}+\mu=0,$ where $\mu>\lambda>0,$ is

• Correct answer
  A

  $\sqrt{\mu-\lambda}$
• B

  $\sqrt{\mu+\lambda}$
• C

  $\sqrt{\mu^{2}-\lambda^{2}}$
• D

  $\mu+\lambda$

Solution

Let the radius of the first circle be $\mathrm{CT}=\mathrm{r}_{1}$. Also, let the radius of the second circle be $\mathrm{CP}=\mathrm{r}_{2}$ In the triangle $\mathrm{PCT}, \mathrm{T}$ is a right angle
$\begin{aligned} \text { So, } P T &=\sqrt{P C^{2}-C T^{2}}=\sqrt{r_{1}^{2}-r_{2}^{2}} \\ &=\sqrt{\left(f^{2}-\lambda\right)-\left(f^{2}-\mu\right)}=\sqrt{\mu-\lambda} \end{aligned}$

Hence, the correct answer is option: 'A'.