Equation of Tangent to a Circle

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

Let the equation of the circle be

x² + y² + 2gx + 2fy + c = 0

Let P(x₁, y₁) 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 (x₁, y₁) is

xx₁ + yy₁ + g(x+x₁) + f(y+y₁) + c = 0

The equation of the tangent at (x₁, y₁) to the circle x² + y² = a² is

xx₁ + yy₁ = a²