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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

SKILLS Plus

Ancillary Circle The ancillary circle of an ellipse is the circle with radius equal to half the length of the minor axis and center the same as the ellipse (see the figure). The ancillary circle is thus the largest circle that can fit within an ellipse.

(a) Find an equation for the ancillary circle of the ellipse x 2 + 4 y 2 = 16 .

(b) For the ellipse and ancillary circle of part (a), show that if ( s , t ) is a point on the ancillary circle, then ( 2 s , t ) is a point on the ellipse.

Chapter 12.2, Problem 61E, SKILLS Plus Ancillary Circle The ancillary circle of an ellipse is the circle with radius equal to

To determine

(a)

The equation for the ancillary circle of the ellipse.

Explanation

Approach:

The basic equation for an ellipse which is used is given as,

x2a2+y2b2=1

Here, a and b are constant values.

The formula to find major axis is given as,

MajorAxis=2a

The formula to find minor axis is given as,

MinorAxis=2b

The equation for the circle is given as,

x2+y2=r2

Here, r is the radius of the circle.

Given:

The equations of the ellipses is given as,

x2+4y2=16

The ancillary circle with radius equal to half the length of the minor axis and center the same as the ellipse as shown in the figure below,

Calculation:

Consider the equation of the ellipse,

x2+4y2=16

Divide both side of the equation by 16, to find the standard equation of the ellipse.

x216+4y216=1616x216+y24=1

Since from the above equation,

To determine

(b)

Whether (2s,t) is a point on the ellipse, if (s,t) is a point on the ancillary circle.

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