The length of the tangent drawn from any point on the circle \(x^2+y^2+2 g x+2 f y+c_1=0\) to the circle \(x^2+y^2+2 g x+2 f y+c_2=0\) is

• Correct answer
  A

  \(\sqrt{C_2-c_1}\)
• B

  \(\sqrt{c_1^2+c_2^2}\)
• C

  \(c_1+c_2\)
• D

  \(c_1-c_2\)

Solution

Circles are \(\begin{aligned} & C_1 \equiv x^2+y^2+2 g x+2 f y+c_1=0 \\ \text {and } & C_2 \equiv x^2+y^2+2 g x+2 f y+c_2=0 \end{aligned}\) Clearly circles are concentric,
Clearly, length of tangent is \(\begin{aligned} A T & =\sqrt{A O^2-O T^2}=\sqrt{r_1^2-r_2^2} \\ & =\sqrt{\left(g^2+f^2-c_1\right)-\left(g^2+f^2-c_2\right)}=\sqrt{c_2-c_1} \end{aligned}\)

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