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
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An ellipse is a stretched circle.
The longest diameter, $PQ$, is called the major axis. The shortest diameter, $RS$, is called the minor axis. The major and minor axes are always perpendicular.
An ellipse can be defined by the parameters $a$ and $b$, where the major axis has length $2a$ and the minor axis has length $2b$ (or vice versa). An ellipse centred on the origin with its major and minor axes aligned with the x and y axes looks like this:
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The Parametric equation of an ellipse is:
$$ x = a \cos{t}\\ y = b \sin{t} $$
Understanding the equations We know that the equations for a point on the unit circle is:
$$ x = \cos{t}\\ y = \sin{t} $$
Multiplying the $x$ formula by $a$ stretches the shape in the x direction, so that is the required width (crossing the x axis at $x=\pm a$:
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