CircleHardBloom L3
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 the point $(g,f)$ is:
Options
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.$\frac{\sqrt{g^{2} + f^{2} + c}}{2\sqrt{g^{2} + f^{2}}}$
Solution
[Key Concept: Distance between two parallel chords of contact = difference of their distances from the centre.]
Equation of chord of contact of the pair of tangent from $(0,0)$ and $(g,f)$ are
$$\begin{array}{r} gx + fy + c = 0\#(1) \end{array}$$
and $gx + fy + \frac{g^{2} + f^{2} + c}{2} = 0$
These lines are parallel
Hence distance $= \left| \frac{c - \frac{g^{2} + f^{2} + c}{2}}{\sqrt{g^{2} + f^{2}}} \right|$
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