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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Coming soon.
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This section looks at the effect of changing the parameter $a$ in the Cartesian equation of the parabola $y^2 = 4 a x$.
Changing the value of $a$ moves the position of the focus and the directrix, which in turn changes the curve. The smaller the value of $a$, the closer the focus and directrix are to the origin.
Use the buttons to set the value of $a$ to different values, and see the effect on the curve.
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We have previously seen how a parabola is defined in terms of parametric equations or alternatively in Cartesian form.
An alternative way to define a parabola is as a locus of points.
Focus and directrix The locus defining a parabola depends on a focus and a directrix.
The focus is a point. For a standard parabola, the focus is located on the x axis a distance $a$ from the origin, that is at the point (a, 0).
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A simple Lissajous figure can be created using the parametric equations:
$$ \begin{align} x = sin(a t)\\ y = sin(b t) \end{align} $$
This animated graph shows how $x$ and $y$ vary with $t$ to create the curve. We are using $a = 3$ and $b = 2$:
The curve is cyclical. As $t$ varies from 0 to 2π radians, the curve traces a complete cycle and returns to the start point (0, 0).
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There are many other interesting parametric curves, in addition to parabolas, hyperbolas and ellipses. Some are show in this section.
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We can convert the parametric equation of a hyperbola into a Cartesian equation (one involving only $x$ and $y$ but not $t$). Here are the parametric equations:
$$ \begin{align} x = c t\\ y = \frac{c}{t} \end{align} $$
We can eliminate $t$ from these equations simply by multiplying $x$ and $y$:
$$ \begin{align} x y &= c t \times \frac{c}{t}\\ x y &= \frac{c^2 t}{t}\\ x y &= c^2 \end{align} $$
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A rectangular hyperbola has the parametric equations:
$$ \begin{align} x = c t\\ y = \frac{c}{t} \end{align} $$
Where $c$ is a positive constant, and $t$ is the independent variable.
We can plot this curve by calculating the values of $x$ and $y$ for various values of $t$, and drawing a smooth curve through them.
Curve for c = 1 Assuming $a = 1$, the parametric equations simplify to:
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