# Equations of Tangent and Normal to the Circle

The equations of tangent and normal to the circle ${x^2} + {y^2} + 2gx + 2fy + c = 0$ at the point $\left( {{x_1},{y_1}} \right)$ are defined by $x{x_1} + y{y_1} + g\left( {x + {x_1}} \right) + f\left( {y + {y_1}} \right) + c = 0$ and $\left( {y - {y_1}} \right)\left( {{x_1} + g} \right) = \left( {x - {x_1}} \right)\left( {{y_1} + f} \right)$ respectively.

Equation of Tangent to the Circle:
The given equation of a circle is

Since the given point lies on the circle, it must satisfy (i). We have

Differentiating both sides of (i) of circle with respect to $x$, we have

If $m$ is the slope of the tangent at $\left( {{x_1},{y_1}} \right)$, then

The equation of the tangent to the circle (i) at the point $\left( {{x_1},{y_1}} \right)$ is

Adding $g{x_1} + f{y_1} + c$ to both sides, we have

This is the equation of the tangent to the circle (i) at point $\left( {{x_1},{y_1}} \right)$.

Equation of Normal to the Circle:

The slope of normal at point $\left( {{x_1},{y_1}} \right)$ is

The equation of normal at $\left( {{x_1},{y_1}} \right)$ is

This is the equation of normal to the circle (i) at point $\left( {{x_1},{y_1}} \right)$.