A tangent to a circle is a straight line which intersects (touches) the circle inexactly one point.

Let the equation of the circle be

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

Let P(x_{1}, y_{1}) be a given point on it.

Let PT be the tangent at P.

The centre of the circle is C(− g, − f).

**How To Derive?** Find slope of the CP. Since CP is perpendicular to PT, find slope of PT. Find equation of tangent in slope-intercept form.

The equation of the tangent at (x_{1}, y_{1}) is

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

The equation of the tangent at (x_{1}, y_{1}) to the circle x^{2} + y^{2} = a^{2} is

**xx _{1} + yy_{1} = a^{2}**