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

{x^2} + {y^2} + 2gx + 2fy + c = 0\,\,\,{\text{ - - - }}\left( {\text{i}} \right)

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

{x_1}^2 + {y_1}^2 + 2g{x_1} + 2f{y_1} + c = 0\,\,\,{\text{ - - - }}\left( {{\text{ii}}} \right)

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

\begin{gathered} 2x + 2y\frac{{dy}}{{dx}} + 2g + 2f\frac{{dy}}{{dx}} + 0 = 0 \\ \Rightarrow x + y\frac{{dy}}{{dx}} + g + f\frac{{dy}}{{dx}} = 0 \\ \Rightarrow \left( {x + y} \right)\frac{{dy}}{{dx}} = - x - g \\ \Rightarrow \frac{{dy}}{{dx}} = - \frac{{x + g}}{{y + f}} \\ \end{gathered}

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

m = {\frac{{dy}}{{dx}}_{\left( {{x_1},{y_1}} \right)}} = - \frac{{{x_1} + g}}{{{y_1} + f}}

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

\begin{gathered} y - {y_1} = m\left( {x - {x_1}} \right) \\ \Rightarrow y - {y_1} = - \frac{{{x_1} + g}}{{{y_1} + f}}\left( {x - {x_1}} \right) \\ \Rightarrow \left( {y - {y_1}} \right)\left( {{y_1} + f} \right) = - \left( {x - {x_1}} \right)\left( {{x_1} + g} \right) \\ \Rightarrow y\left( {{y_1} + f} \right) - {y_1}\left( {{y_1} + f} \right) = - x\left( {{x_1} + g} \right) + {x_1}\left( {{x_1} + g} \right) \\ \Rightarrow y{y_1} + fy - {y_1}^2 - f{y_1} = - x{x_1} - gx + {x_1}^2 + g{x_1} \\ \Rightarrow x{x_1} + y{y_1} + gx + fy = {x_1}^2 + {y_1}^2 + g{x_1} + f{y_1} \\ \end{gathered}

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

\begin{gathered} x{x_1} + y{y_1} + gx + fy + g{x_1} + f{y_1} + c = {x_1}^2 + {y_1}^2 + g{x_1} + f{y_1} + g{x_1} + f{y_1} + c \\ \Rightarrow x{x_1} + y{y_1} + g\left( {x + {x_1}} \right) + f\left( {y + {y_1}} \right) + c = {x_1}^2 + {y_1}^2 + 2g{x_1} + 2g{y_1} + c \\ \Rightarrow x{x_1} + y{y_1} + g\left( {x + {x_1}} \right) + f\left( {y + {y_1}} \right) + c = 0 \\ \end{gathered}

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

 - \frac{1}{m} = \frac{{{y_1} + f}}{{{x_1} + g}}

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

\begin{gathered} y - {y_1} = \frac{{{y_1} + f}}{{{x_1} + g}}\left( {x - {x_1}} \right) \\ \Rightarrow \left( {y - {y_1}} \right)\left( {{x_1} + g} \right) = \left( {x - {x_1}} \right)\left( {{y_1} + f} \right) \\ \end{gathered}

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