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Calculus / Calculus: Early Transcendentals 1 / Chapter 10.4 / Problem 31E

Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

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Graphing ellipses Sketch the graph of the

Chapter 9, Problem 31E

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QUESTION:

Sketch the graph of the following ellipses. Plot and label the coordinates of the vertices and foci, and find the lengths of the major and minor axes. Use a graphing utility to check your work.

\(\frac{x^2}{5}+\frac{y^2}{7}=1\)

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QUESTION:

Sketch the graph of the following ellipses. Plot and label the coordinates of the vertices and foci, and find the lengths of the major and minor axes. Use a graphing utility to check your work.

\(\frac{x^2}{5}+\frac{y^2}{7}=1\)

ANSWER:

Solution 31E

Step 1:

In this problem we need to Plot and label the coordinates of the vertices and foci of the ellipse $\frac{{x}^{2}}{5}+\frac{{y}^{2}}{7}=1$ and we have to find the lengths of the major and minor axes.

The standard equation of the ellipse with center (0,0) and major axis on the y-axis is $\frac{{x}^{2}}{{b}^{2}}+\frac{{y}^{2}}{{a}^{2}}=1$ where a>b

Here we have $\frac{{x}^{2}}{5}+\frac{{y}^{2}}{7}=1$

${b}^{2}=5,\ {a}^{2}=7$

$\Rightarrow{}a=\sqrt{7},b=\sqrt{5}$

Vertices of the ellipse: $(0,\pm{}a)$

Therefore the coordinates of the ellipse $\frac{{x}^{2}}{5}+\frac{{y}^{2}}{7}=1$ is $(0,\pm{}\sqrt{7})$

Foci of the ellipse: $(0,\pm{}c)$ where ${c}^{2}={a}^{2}-{b}^{2}$

$\Rightarrow{}{c}^{2}=7-5=2$

$\Rightarrow{}{c}^{2}=2$

$\Rightarrow{}c=\sqrt{2}$

Therefore the foci of the ellipse $\frac{{x}^{2}}{4}+\frac{{y}^{2}}{16}=1$ is $(0,\pm{}\sqrt{2})$

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