Solved: Conjecture (a) Show that the equation of an ellipse can be written as (b) Use a

Chapter 10, Problem 82

(choose chapter or problem)

(a) Show that the equation of an ellipse can be written as

\(\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{a^{2}\left(1-e^{2}\right)}=1\).

(b) Use a graphing utility to graph the ellipse

\(\frac{(x-2)^{2}}{4}+\frac{(y-3)^{2}}{4\left(1-e^{2}\right)}=1\) 

for  e = 0. 95, e = 0. 75, e = 0. 5, e = 0. 25, and  e = 0.

(c) Use the results of part (b) to make a conjecture about the change in the shape of the ellipse as  \(e\)  approaches 0.

Text Transcription:

(x - h)^2 / a^2 + (y - k)^2 / a^2 (1 - e^2) = 1

(x - 2)^2 / 4 + (y - 3)^2 / 4(1 - e^2) = 1

e  

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