Problems on Kepler's First Law
Problem 1.1:
Calculate the eccentricity of an ellipse with one focus at the origin and the
other at
(- 2k, 0), and semimajor axis length
3k.
[Solution]
Problem 1.2:
For an ellipse with its major axis parallel to the
x-direction and its
rightmost focus at the origin, derive the position of the other focus in terms
of its eccentricity
ε and
k, where
k is defined as
k = a(1 - ε2).
[Solution]
Problem 1.3:
The general equation for orbital motion is given by:
Where the
k is the same
k as in the last problem:
k = a(1 - ε2) = . Show that when
ε = 0, this reduces to an equation
for a circle. What is the radius of this circle?
[Solution]
Problem 1.4:
Derive the formula for the area of an ellipse by integration.
[Solution]
Problem 1.5:
Prove that for a point on an ellipse, the sum of the distances to each foci is a
constant.
[Solution]
. Show that when ε = 0, this reduces to an equation
for a circle. What is the radius of this circle?
[Solution]
