Equation of Chord of Contact of Tangents
The general equation of the circle is
x2 + y2 + 2gx + 2fy + c = 0
Let P(x1, y1) be a point outside the circle. Let the tangents from P(x1, y1) touch the circle at Q(x2, y2) and R(x3, y3).
The equation of the tangent PQ at Q (x2, y2) is
xx2 + yy2 + g(x + x2) + f(y + y2) + c = 0
The equation of the tangent PR at R(x3, y3) is
xx3 + yy3 + g(x + x3) + f(y + y3) + c = 0
The point (x1, y1) satisfy both the equations of tangent.
x1x2 + y1y2 + g(x1 + x2) + f(y1 + y2) + c = 0
x1x3 + y1y3 + g(x1 + x3) + f(y1 + y3) + c = 0
These equations show that (x2, y2) and (x3, y3) lie on the line
xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0
The straight line xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0 represents the equation of QR, chord of contact of tangents from (x1, y1).