Question

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

• A. $\frac{2(g^2+f^2-c)}{\sqrt{g^2+f^2}}$
• B. $\frac{g^2+f^2-c}{\sqrt{g^2+f^2}}$
• C. $\frac{g^2+f^2-c}{2\sqrt{g^2+f^2}}$
• D. none of these

Answer 1

Answer: (C)

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

Equation of $AB:gx+fy+c=\mathrm{0.........}\left(i\right)$

Equation of $CD:gx+fy+g(x+g)+f(y+f)+c=0$

$gx+fy+\frac{g^2+f^2+c}{2}=\mathrm{0.........}\left(ii\right)$

Distance between $AB$ and $CD$ will be

$\Rightarrow \left|\frac{\frac{g^2+f^2-c}{2}}{\sqrt{g^2+f^2}}\right|=\frac{g^2+f^2-c}{2\sqrt{g^2+f^2}}$