?A point on an ellipse with major axis length 2a and minor axis length 2b has the

Chapter 5, Problem 203

(choose chapter or problem)

A point on an ellipse with major axis length 2a and minor axis length 2b has the coordinates \((a \cos \theta, b \sin \theta)\), \(0 \leq \theta \leq 2 \pi\).

(a) Show that the distance from this point to the focus at (-c, 0) is \(d(\theta)=a+c \cos \theta\), where \(c=\sqrt{a^{2}-b^{2}}\)

(b) Use these coordinates to show that the average distance \(\bar{d}\) from a point on the ellipse to the focus at (-c, 0), with respect to angle \(\theta\), is a.

Text Transcription:

(a cos theta, b sin theta)

0 leq theta leq 2 pi

d(theta)=a+c cos theta

c=sqrt a^2-b^2

bar d

theta