   Chapter 10.1, Problem 78E

Chapter
Section
Textbook Problem

ConjectureShow that the equation of an ellipse can be written as ( x − h ) 2 a 2 + ( y − k ) 2 a 2 ( 1 − e 2 ) = 1 .(b) Use a graphing utility to graph the ellipse ( x − 2 ) 2 4 + ( y − 3 ) 2 4 ( 1 − e 2 ) = 1 for e = 0.95 , e = 0.75 , e = 0.5 , e = 0.25 ,   and   e = 0 Use the results of part (b) to make a conjecture about the change in the shape of the ellipse as e approaches 0.

(a)

To determine

To prove: The equation of an ellipse can be re-written in the form of as (xh)2a2+(yk)2a2(1e2)=1.

Explanation

Given:

The equation, (xh)2a2+(yk)2a2(1e2)=1.

Formula used:

The standard equation of an ellipse is, (xh)2a2+(yk)2b2=1 and eccentricity of the ellipse formulated by the equation e=a2b2a2.

Proof:

Consider the standard equation of the ellipse, (xh)2a2+(yk)2b2=1.

Since, the eccentricity of the ellipse is formulated as, e2=a2b2a2

(b)

To determine

To graph: The ellipse, (x2)24+(y3)24(1e2)=1 for different values of e with the use of graphing utility.

(c)

To determine
The conjecture of the change in shape of the graph of the ellipse in part (b) as e approaches 0.

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