# Equation of Chord of Contact of Tangents

The general equation of the circle is

x^{2} + y^{2} + 2gx + 2fy + c = 0

Let P(x_{1}, y_{1}) be a point outside the circle. Let the tangents from P(x_{1}, y_{1}) touch the circle at Q(x_{2}, y_{2}) and R(x_{3}, y_{3}).

The equation of the tangent PQ at Q (x_{2}, y_{2}) is

xx_{2} + yy_{2} + g(x + x_{2}) + f(y + y_{2}) + c = 0

The equation of the tangent PR at R(x_{3}, y_{3}) is

xx_{3} + yy_{3} + g(x + x_{3}) + f(y + y_{3}) + c = 0

The point (x_{1}, y_{1}) satisfy both the equations of tangent.

x_{1}x_{2} + y_{1}y_{2} + g(x_{1} + x_{2}) + f(y_{1} + y_{2}) + c = 0

x_{1}x_{3} + y_{1}y_{3} + g(x_{1} + x_{3}) + f(y_{1} + y_{3}) + c = 0

These equations show that (x_{2}, y_{2}) and (x_{3}, y_{3}) lie on the line

xx_{1} + yy_{1} + g(x + x_{1}) + f(y + y_{1}) + c = 0

The straight line **xx _{1} + yy_{1} + g(x + x_{1}) + f(y + y_{1}) + c = 0** represents the equation of QR, chord of contact of tangents from (x

_{1}, y

_{1}).