Question

The distance between the chords of contact of tangents to the circle $x^2+y^2+2gx+2fy+c=0$ from the origin and from the point $(g,f)$ is

• A. $\sqrt{g^2+f^2}$
• B. $\frac{\sqrt{9^2+f^2-c}}{2}$
• C. $\frac{g^2+f^2-c}{2\sqrt{g^2+f^2}}$
• D. $\frac{\sqrt{9^2+f^2+c}}{2\sqrt{s^2+f^2}}$

Answer 1

The correct option is C $\frac{g^2+f^2-c}{2\sqrt{g^2+f^2}}$

given equation of circle,

$x^2+y^2+2gx+2fy+c=0$

Tangent to this circle at $(p,q)$

$xp+yq+g(x+p)+f(y+q)+c=0$...(1)

equations of chord of contact of tangents from origin $(0,0)$ and point $(g,f)$ given circle are,

$x\left(0\right)+y\left(0\right)+g(x+0)+f(y+0)+c=0$

$\Rightarrow gx+fy+c=0$...(2)

$xg+yf+g(x+g)+f(y+f)+c=0$

$\Rightarrow gx+fy+\frac{1}{2}(g^2+f^2+c)=0$...(3)

From (2) and (3) it's clear that, equations are parallel.

Distance between these chords is given by,

$\frac{\frac{1}{2}(g^2+f^2+c)-c}{\sqrt{g^2+f^2}}$

$\therefore \frac{g^2+f^2-c}{2\sqrt{g^2+f^2}}$ is the distance