Consider an ellipse whose semi-major axis is and semi-minor axis is

MATHEMATICS · Multi-part · AP · AP Precalculus · Functions with Parameters, Vectors, and Matrices · Functions with Parameters, Vectors, and Matrices

Problem

Consider an ellipse whose semi-major axis is

a

and semi-minor axis is

b

. Answer the questions below. (i) Establish an expression for the eccentricity

e

entirely in terms of

a

and

b

. (ii) Using your expression, explain why an ellipse with

b

nearly equal to

a

appears almost circular. (iii) Show that

e<0.5

if and only if

\dfrac{b}{a}>\dfrac{\sqrt{3}}{2}

and interpret this condition geometrically.