# Find the Equation of the Tangent Line to the Ellipse

Find the equation of the tangent and normal to the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ at the point $\left( {a\cos \theta ,b\sin \theta } \right)$.

We have the standard equation of an ellipse

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

Let $m$ be the slope of the tangent at the given point $\left( {a\cos \theta ,b\sin \theta } \right)$, then

The equation of the tangent at the given point $\left( {a\cos \theta ,b\sin \theta } \right)$ is

This is the equation of the tangent to the given ellipse at $\left( {a\cos \theta ,b\sin \theta } \right)$.

The slope of the normal at $P\left( {a\cos \theta ,b\sin \theta } \right)$ is $- \frac{1}{m} = - \left( { - \frac{{a\sin \theta }}{{b\cos \theta }}} \right) = \frac{{a\sin \theta }}{{b\cos \theta }}$

The equation of the normal at the point $P\left( {a\cos \theta ,b\sin \theta } \right)$ is

This is the equation of the normal to the given ellipse at $P\left( {a\cos \theta ,b\sin \theta } \right)$.