GraphicMaths

Visualising maths

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 $x = -a$.