## Cartesian equation of ellipse

The Cartesian equation of an ellipse is:

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

## Understanding the equation

Starting from the parametric equations:

```
$$
x = a \cos{t}\\
y = b \sin{t}
$$
```

These can be rearranged to give:

```
$$
\cos{t} = \frac{x}{a}\\
\sin{t} = \frac{y}{b}
$$
```

We also know that:

```
$$
\cos^2{t} + \sin^2{t} = 1
$$
```

This follows from Pythagoras.

Substituting for `$\cos{t}$`

and `$\sin{t}$`

gives

```
$$
\left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = 1\\
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
$$
```

## Intersection with x and y axes

We can check that this equation correctly predicts where the curve intersects the x and y axes. For the y axis, `$x = 0$`

:

```
$$
\begin{align}
\frac{y^2}{b^2} = 1\\
y^2 = b^2\\
y = \pm b
\end{align}
$$
```

Which is correct, the curve intersect the y axis at `$y = b$`

and `$y = -b$`

.

For the x axis, `$y = 0$`

:

```
$$
\begin{align}
\frac{x^2}{a^2} = 1\\
x^2 = a^2\\
x = \pm a
\end{align}
$$
```

Which is also correct, the curve intersect the x axis at `$x = a$`

and `$x = -a$`

.