# Equation of Tangent and Normal in Different Forms for a ellipse

### Equation of Tangent to an Ellipse:

An ellipse is a curve on a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve. The general equation of an ellipse with the center at the origin is:

$\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$

Where $a$ and $b$ are the semi-major and semi-minor axes, respectively.

#### Equation of Tangent:

The equation of the tangent to an ellipse at a point $\left({x}_{1},{y}_{1}\right)$ on the ellipse is given by:

$\frac{x{x}_{1}}{{a}^{2}}+\frac{y{y}_{1}}{{b}^{2}}=1$

Where $a$ and $b$ are the semi-major and semi-minor axes of the ellipse, and $\left({x}_{1},{y}_{1}\right)$ is the point of tangency.

### Equation of Normal to an Ellipse:

The normal line to an ellipse at a point $\left({x}_{1},{y}_{1}\right)$ is perpendicular to the tangent at that point. The equation of the normal line to an ellipse at the point $\left({x}_{1},{y}_{1}\right)$ is given by:

$\frac{y{x}_{1}}{{a}^{2}}-\frac{x{y}_{1}}{{b}^{2}}=\frac{{a}^{2}-{b}^{2}}{ab}$

### Parametric Form of Tangent and Normal:

For an ellipse with parametric equations:

$x=a\mathrm{cos}\left(\theta \right)$

$y=b\mathrm{sin}\left(\theta \right)$

The parametric equations of the tangent and normal lines to the ellipse at $\left({x}_{1},{y}_{1}\right)$ are:

Tangent:

$x\cdot \frac{{x}_{1}}{{a}^{2}}+y\cdot \frac{{y}_{1}}{{b}^{2}}=1$

Normal:

$y\cdot \frac{{x}_{1}}{{a}^{2}}-x\cdot \frac{{y}_{1}}{{b}^{2}}=\frac{{a}^{2}-{b}^{2}}{ab}$

### Polar Form of Tangent and Normal:

For an ellipse with polar coordinates:

$r=\frac{ab}{\sqrt{\left(b\mathrm{cos}\left(\theta \right){\right)}^{2}+\left(a\mathrm{sin}\left(\theta \right){\right)}^{2}}}$

The equations of the tangent and normal lines to the ellipse at $\theta ={\theta }_{0}$ are:

Tangent (Polar Form):

$r\cdot {r}_{0}=ab$

Normal (Polar Form):

$r\cdot {r}_{0}=\frac{{a}^{2}-{b}^{2}}{ab}$

### Example: Finding Equations of Tangent and Normal Lines

Let's consider an ellipse with the equation:

$\frac{{x}^{2}}{9}+\frac{{y}^{2}}{4}=1$

This ellipse has a semi-major axis of length $a=3$ and a semi-minor axis of length $b=2$.

Let's determine the equations of the tangent and normal lines to the given ellipse at the point $\left(3\mathrm{cos}\left(\theta \right),2\mathrm{sin}\left(\theta \right)\right)$ where $\theta =\frac{\pi }{6}$ in different forms.

#### Parametric Form:

For the given ellipse with parametric equations:

$x=3\mathrm{cos}\left(\theta \right)$

$y=2\mathrm{sin}\left(\theta \right)$

##### Tangent Line (Parametric Form):

Using the parametric equation of the tangent line:

$x\cdot \frac{{x}_{1}}{{a}^{2}}+y\cdot \frac{{y}_{1}}{{b}^{2}}=1$

Substitute $x=3\mathrm{cos}\left(\theta \right)$, $y=2\mathrm{sin}\left(\theta \right)$, ${x}_{1}=3\mathrm{cos}\left(\frac{\pi }{6}\right)=\frac{3\sqrt{3}}{2}$, ${y}_{1}=2\mathrm{sin}\left(\frac{\pi }{6}\right)=1$:

$3\mathrm{cos}\left(\theta \right)\cdot \frac{3\sqrt{3}}{9}+2\mathrm{sin}\left(\theta \right)\cdot \frac{1}{4}=1$

Simplify this equation to get the equation of the tangent line in parametric form.

##### Normal Line (Parametric Form):

Using the equation for the normal line:

$y\cdot \frac{{x}_{1}}{{a}^{2}}-x\cdot \frac{{y}_{1}}{{b}^{2}}=\frac{{a}^{2}-{b}^{2}}{ab}$

Substitute the values and simplify to find the equation of the normal line in parametric form.

#### Polar Form:

For the given ellipse with polar equation:

$r=\frac{6}{\sqrt{\left(2\mathrm{cos}\left(\theta \right){\right)}^{2}+\left(3\mathrm{sin}\left(\theta \right){\right)}^{2}}}$

##### Tangent Line (Polar Form):

The equation of the tangent line in polar form is $r\cdot {r}_{0}=ab$. Substitute the given polar equation to find the equation of the tangent line in polar form.

##### Normal Line (Polar Form):

The equation of the normal line in polar form is $r\cdot {r}_{0}=\frac{{a}^{2}-{b}^{2}}{ab}$. Substitute the given polar equation to find the equation of the normal line in polar form.

### Example:

Let's consider an ellipse with the equation:

$\frac{{x}^{2}}{9}+\frac{{y}^{2}}{4}=1$

This ellipse has a semi-major axis $a=3$ and a semi-minor axis $b=2$.

Find the equations of the tangent and normal lines to the ellipse at the point $\left(3\sqrt{2},2\right)$.

#### Equation of Tangent:

The equation of the tangent to an ellipse at a point $\left({x}_{1},{y}_{1}\right)$ on the ellipse is given by:

$\frac{x{x}_{1}}{{a}^{2}}+\frac{y{y}_{1}}{{b}^{2}}=1$

At the point $\left(3\sqrt{2},2\right)$, the equation of the tangent line is:

$\frac{x\cdot 3\sqrt{2}}{9}+\frac{y\cdot 2}{4}=1$

Simplify to get the equation of the tangent line in general form.

#### Equation of Normal:

The equation of the normal line to an ellipse at a point $\left({x}_{1},{y}_{1}\right)$ is given by:

$\frac{y{x}_{1}}{{a}^{2}}-\frac{x{y}_{1}}{{b}^{2}}=\frac{{a}^{2}-{b}^{2}}{ab}$

At the point $\left(3\sqrt{2},2\right)$, the equation of the normal line is:

$\frac{y\cdot 3\sqrt{2}}{9}-\frac{x\cdot 2}{4}=\frac{9-4}{6}$

Simplify to get the equation of the normal line in general form.

### Parametric Form of Tangent and Normal:

For an ellipse with parametric equations:

$x=a\mathrm{cos}\left(\theta \right)$

$y=b\mathrm{sin}\left(\theta \right)$

At the point $\left(3\sqrt{2},2\right)$, we have $a=3$ and $b=2$. So, ${x}_{1}=3\sqrt{2}$ and ${y}_{1}=2$.

#### Tangent (Parametric Form):

The equation of the tangent line using parametric form is:

$x\cdot \frac{3\sqrt{2}}{9}+y\cdot \frac{2}{4}=1$Simplify to get the equation of the tangent line in general form.

#### Normal (Parametric Form):

The equation of the normal line using parametric form is:

$y\cdot \frac{3\sqrt{2}}{9}-x\cdot \frac{2}{4}=\frac{9-4}{6}$

Simplify to get the equation of the normal line in general form.

These equations in different forms illustrate the tangent and normal lines to the given ellipse at the point $\left(3\sqrt{2},2\right)$.