# Equation of Tangent and Normal to the Ellipse

The equations of tangent and normal to the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ at the point $\left( {{x_1},{y_1}} \right)$ are $\frac{{{x_1}x}}{{{a^2}}} + \frac{{{y_1}y}}{{{b^2}}} = 1$ and ${a^2}{y_1}x - {b^2}{x_1}y - \left( {{a^2} - {b^2}} \right){x_1}{y_1} = 0$ respectively.

Consider that the standard equation of ellipse with vertex at origin $\left( {0,0} \right)$ can be written as

Since the point $\left( {{x_1},{y_1}} \right)$ lies on the given ellipse, it must satisfy equation (i). So we have

Now differentiating equation (i) on both sides with respect to $x$, we have

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

The equation of a tangent at the given point $\left( {{x_1},{y_1}} \right)$ is

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

The slope of the normal at $\left( {{x_1},{y_1}} \right)$ is $- \frac{1}{m} = - \left( { - \frac{{{a^2}{x_1}}}{{{b^2}{y_1}}}} \right) = \frac{{{a^2}{x_1}}}{{{b^2}{y_1}}}$

The equation of the normal at the point $\left( {{x_1},{y_1}} \right)$ is $y - {y_1} = \frac{{{a^2}{x_1}}}{{{b^2}{y_1}}}\left( {x - {x_1}} \right)$

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