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Q13.

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Precalculus Mathematics for Calculus
Found in: Page 797
Precalculus Mathematics for Calculus

Precalculus Mathematics for Calculus

Book edition 7th Edition
Author(s) James Stewart, Lothar Redlin, Saleem Watson
Pages 948 pages
ISBN 9781337067508

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Short Answer

Graphing Ellipses An equation of an ellipse is given. (a) Find the vertices, foci, and eccentricity of the ellipse.

(b) Determine the lengths of the major and minor axes. (c) Sketch a graph of the ellipse.

x249+y225=1

(a).The vertices, foci and eccentricity of the ellipse x249+y225=1 are: (±7,0), (±26,0) and 267 respectively.

(b). The length of the major axis is 14 and the length of the minor axis is 10 for the ellipse x249+y225=1.

(c). Graph of the ellipse x249+y225=1 is as follows:

See the step by step solution

Step by Step Solution

a.Step 1. Given information.

The equation of the ellipse, x249+y225=1

Step 2. The vertices, foci, and eccentricity of the ellipse.

If we compare the given equation x249+y225=1 with standard form

(x-h)2a2+(y-k)2b2=1.

This gives us

h=0,k=0a2=49,a=7b2=25.b=5

So vertices of the ellipse are (±a,0)or(±7,0)

Foci are \[(\pm c,0)\],where c2=a2-b2

c=a2-b2

By substituting values we will get c2=49-25

Therefore Foci are (±26,0)

Eccentricity e=ca. We have values of both, by substituting themwe get e=267

b.Step 1. Given information.

The equation of the ellipse, x249+y225=1

Step 2. The lengths of major and minor axes.

The equation of the given ellipse is x249+y225=1

By comparing the given equation with the general form we found that

a=7b=5

Length of Major axis =2a

=2(7)=14

Length of Minor axis=2b

=2(5)=10

c.Step 1. Given information.

The equation of the given ellipse is x249+y225=1.

Step 2. We have to solve the given equation for y.

From the equation of ellipse, we can see that denominator of x2 is greater than that of y2. So this Would be a Horizontal ellipse. By comparing the given equation with the standard form equation (x-h)2a2+(y-k)2b2=1

x249+y225=1

We will get:

h=0k=0a2=49a=7b2=25b=5c=a2b2c=24c=26

Foci of the ellipse (±c,0) are (±26,0), vertices (±a,0) are (±7,0).

The graph will be

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