1CParamEllipses


 * Parametric Equations of Ellipses**

We can use parametric equations to define an ellipse. math \\ . \qquad x = a \cos (t) \\. \\ . \qquad y = b \sin (t) math

where
 * a is the radius in the x direction
 * b is the radius in the y direction

math \\ \text {if we use radians, then } t \in \lbrack 0, 2\pi \rbrack \text { (or equivalent)} \\. \\ \text {if we use degrees, then } t \in \lbrack 0, 360^\circ \rbrack math

math \\ \text {Domain: } x \in \lbrack -a, \, a \rbrack \\. \\ \text {Range: } \: y \in \lbrack -b, \, b \rbrack math

Note: if t is set to a domain smaller than 2 p, a partial ellipse will be obtained.

We can confirm that these parametric equations create an ellipse by finding the cartesian equation from the equations. This can be done by isolating the trig functions, squaring each equation, then adding the two equations.

math \\ . \qquad x=a \, \cos(t) \qquad \qquad \qquad y = b \, \sin(t) \\. \\ . \qquad \cos(t)=\dfrac{x}{a} \qquad \qquad \qquad \; \sin(t)=\dfrac{y}{b} \\. \\ . \qquad \cos^2(t)=\dfrac{x^2}{a^2} \qquad \qquad \quad \;\; \sin^2(t)=\dfrac{y^2}{b^2} math

Add the equations: math . \qquad \cos^2(t)+\sin^2(t)=\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2} math

Using Pythagoras, we get math . \qquad \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 math

This is the equation of an ellipse with its centre at the origin.

General Parametric Equations of Ellipses

For an ellipse that has been translated h units to the right and k units up, the parametric equations become: math \\ . \qquad x = a \cos (t) + h \\. \\ . \qquad y = b \sin (t) + k math

Center at (h, k)

math \\ \text {if we use radians, then } t \in \lbrack 0, 2\pi \rbrack \text { (or equivalent)} \\ .\\ \text {if we use degrees, then } t \in \lbrack 0, 360^\circ \rbrack math

math \\ \text {Domain: } x \in \lbrack h-a, \, h+a \rbrack \\. \\ \text {Range: } \: y \in \lbrack k-b, \, k+b \rbrack math

Example

Find the cartesian equation of the graph with parametric equations:

math \\ . \qquad x = 3 \cos (t) - 2 \quad \text { and} \\. \\ . \qquad y = 5 \sin (t) + 1 \\. \\ . \qquad \lbrace t \in \lbrack 0, 2\pi \rbrack \rbrace math

This can be done by isolating the trig functions, squaring each equation, then adding the two equations.

math \\ . \qquad x=3\cos(t)-2 \qquad \qquad \qquad y=5\sin(t)+1 \\. \\ . \qquad 3\cos(t)=x+2 \qquad \qquad \qquad 5\sin(t)=y-1 math . math \\ . \qquad \cos(t)=\dfrac{x+2}{3} \qquad \qquad \qquad \; \sin(t) = \dfrac{y-1}{5} \\. \\ . \qquad \cos^2(t)=\dfrac{(x+2)^2}{9} \qquad \qquad \;\;\; \sin^2(t)=\dfrac{(y-1)^2}{25} math

Add equations math . \qquad \cos^2(t)+\sin^2(t)=\dfrac{(x+2)^2}{9}+\dfrac{(y-1)^2}{25} math

Using Pythagoras, this becomes math . \qquad \dfrac{(x+2)^2}{9}+\dfrac{(y-1)^2}{25}=1 math

This is the cartesian equation of an ellipse with centre at (–2, 1) and with a = 3, b = 5

math \\ \text {Domain: } x \in [ -2-3, \, -2+3 ] \; \Rightarrow \; x \in [ -5, \, 1 ] \\. \\ \text {Range: } \:\; y \in [ 1-5, \, 1+5 ] \quad \Rightarrow \quad y \in [-4, \, 6] math

Return to Ellipses

Return to Summary of Conic Sections

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