Question

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_____

• A. $\sqrt{c_{2}-c_{1}}$
• B. $\sqrt{c_{1}^{2}-c_{2}^{2}}$
• C. $c_{1}+c_{2}$
• D. $c_{2}-c_{1}$

Answer 1

Answer: (A)

Circles are
$C_{1} \equiv x^{2}+y^{2}+2 g x+2 f y +c_{1}=0$
and $C_{2} \equiv x^{2}+y^{2}+2 g x+2 f y+ c_{2}=0$
Clearly circles are concentric,

Cleariy, length of tangent is
$AT =\sqrt{ AO ^{2}- OT ^{2}}=\sqrt{r_{1}{ }^{2}-r_{2}{ }^{2}}$
$=\sqrt{\left(g^{2}+f^{2}-c_{1}\right)-\left(g^{2}+f^{2}-c_{2}\right)}$
$=\sqrt{c_{2}-c_{1}}$