Question

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

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

Answer 1

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

The equation of chord of contact is given by $T=0,T=(x{x}_1+y{y}_1+g(x+x_1)+f(y+y_1)+c)$ where $P(x_1,y_1)$ from which the tangents are drawn.

$T_{(0,0)}=0+0+g(x+0)+f(y+0)+c=0$

$⟹gx+fy+c=0$

$⟹2gx+2fy+2c=0–\left(1\right)$

$T_{(g,f)}=gx+fy+g(x+g)+f(y+f)+c=0$

$⟹2gx+2fy+g^2+f^2+c=0–\left(2\right)$

(1)And (2) are parallel lines

Distance between them $=\frac{c_2–c_1}{\sqrt{a^2+b^2}}$

$d=\frac{g^2+f^2+c–2c}{\sqrt{(2g{)}^2+(2f{)}^2}}=\frac{g^2+f^2-c}{2\sqrt{g^2+f^2}}$