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Found in: Page 120

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# Give an example of a parametrization of the ellipse ${{\mathbit{x}}}^{{\mathbf{2}}}{\mathbf{+}}\frac{{\mathbf{y}}^{\mathbf{2}}}{\mathbf{4}}{\mathbf{=}}{\mathbf{1}}$ in ${{\mathbf{ℝ}}}^{{\mathbf{2}}}$ . See Example .

$\left[\begin{array}{c}\mathrm{cos}\left(t\right)\\ 2\mathrm{sin}\left(t\right)\end{array}\right]$ is a parametrization of the ellipse ${x}^{2}+\frac{{y}^{2}}{4}=1$ in ${\mathrm{ℝ}}^{2}$ .

See the step by step solution

## Step 1: Consider the parameters.

The image of a function consists of all the values the function takes in its target space. If f is a function from X to Y, then

$image\left(f\right)=\left\{f\left(x:xinX\right)\right\}\phantom{\rule{0ex}{0ex}}=\left\{binY:b=f\left(x\right),forsomexinX\right\}$

The standard form of an ellipse is, $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$

The parametrization of this ellipse is,

$x=\left(a\right)\mathrm{cos}\left(t\right),y=\left(b\right)\mathrm{sin}\left(t\right)\phantom{\rule{0ex}{0ex}}\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}\left(a\right)\mathrm{cos}\left(t\right)\\ \left(b\right)\mathrm{sin}\left(t\right)\end{array}\right]$

The ellipse equation is, $\frac{{x}^{2}}{1}+\frac{{y}^{2}}{4}=1$

The parametrization of this ellipse is,

$x=\left(1\right)\mathrm{cos}\left(t\right),Y=\left(2\right)\mathrm{sin}\left(t\right)\phantom{\rule{0ex}{0ex}}\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}\left(1\right)\mathrm{cos}\left(t\right)\\ \left(2\right)\mathrm{sin}\left(t\right)\end{array}\right]$

$\left[\begin{array}{c}\mathrm{cos}\left(t\right)\\ 2\mathrm{sin}\left(t\right)\end{array}\right]$ is a parametrization of the ellipse ${x}^{2}+\frac{{y}^{2}}{4}=1$ in ${\mathrm{ℝ}}^{2}$ .