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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336

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Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336
Textbook Problem

Find the vertices and foci of the ellipse and sketch its graph.

11. x 2 2 + y 2 4 = 1

To determine

To Find: The vertices and foci of the ellipse for the equation x22+y24=1 .

Explanation

Given:

The ellipse equation is as follows.

x22+y24=1 (1)

Then, compare the equation (1) with the standard equation of ellipse,

b>a

So, the foci of an ellipse are located on the y axis using equation below,

x2b2+y2a2=1

Calculation:

Substitute the value 2 for b2 and 4 for a2 .

b2=2b=2

a2=4a=4a=2

Compute the vertices:

The vertices is said to be (0,±a)

Substitute the value 2 for a .

Then, the vertices of the ellipse are (0,±2) .

Compute the value of c using the equation below.

c2=a2b2

Substitute the value 2 for b and 2 for a .

c2=22(2)2c2=42c=2

The foci is said to be (0,±c) .

Then, the foci of the ellipse is (0,±2) .

Compute the value of radius using the equation (1).

r=aba2cos2θ+b2sin2θ

Substitute the value 0 for θ , 2 for a and 2 for b in the above equation.

r=2×222cos20+(2)2sin20r=1

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