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Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742

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BuyFindarrow_forward

Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742
Textbook Problem

13-24 ■ Graphing Ellipses An equation of an ellipse is given. (a) Find the center, vertices, and foci of the ellipse. (b) Determine the lengths of the major and minor axes. (c) Sketch a graph of the ellipse.

( x 2 ) 2 25 + ( y + 3 ) 2 16 = 1

To determine

(a)

To find:

The center, vertices, and foci of the ellipse.

(x2)225+(y+3)216=1

Explanation

Given:

The equation of the ellipse is,

(x2)225+(y+3)216=1

Approach:

The following equation is the basic equation for an ellipse that is not centered at the origin.

(xh)2a2+(yk)2b2=1

With the following properties,

Center C=(h,k)……(1)

Vertices V=(±a+h,0+k)……(2)

Foci F=(±c+h,0+k)……(3)

Where

c2=a2b2……(4)

Calculation:

Consider the equation,

(x2)225+(y+3)216=1

Convert the above equation into the standard equation of an ellipse.

(x2)252+(y+3)242=1

Compare the above equation of the ellipse with the standard equation of an ellipse.

a=5b=4h=2k=3

Calculate the center of the ellipse.

Substitute 2 for h and 3 for k in the equation (1).

C=(2,3)

Calculate the vertices of the ellipse.

Substitute 5 for a, 2 for h and 3 for k in the equation (2).

V1=(5+2,03)=(7,3)

Substitute 5 for a, 2 for h and 3 for k in the equation (2)

To determine

(b)

The length of the major and minor axes.

To determine

(c)

To sketch:

The graph of the ellipse.

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