Question

The length of the tangent drawn from any point on the circle $x^2+y^2+2gx+2fy+p=0$ to the circle $x^2+y^2+2gx+2fy+q=0$ is:

• A. $\sqrt{q-p}$
• B. $\sqrt{p-q}$
• C. $\sqrt{q+p}$
• D. $\sqrt{2q+p}$

Answer 1

The correct option is A $\sqrt{q-p}$

$S_1=x^2+y^2+2gx+2fy+p=0$

$S_2=x^2+y^2+2gx+2fy+q=0$

$A{B}^2=O{B}^2-O{A}^2$

$A\left(x_1{y}_1\right)=x_1^2+y_1^2+2g{x}_{1}2f{y}_1+p=0$

$x_1^2+y_1^2+2g{x}_1+2f{y}_1=-p$

length $=\sqrt{x_1^2+y_1^2+2g{x}_1+2g{y}_1+q}$

length $=\sqrt{q-p}$