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 $$\left( {{x_1},{y_1}} \right)$$.