# Kepler and Gravitation

### Problems on Kepler's First Law

Problem : Calculate the eccentricity of an ellipse with one focus at the origin and the other at $(-2k, 0)$, and semimajor axis length $3k$.

Problem : 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 $\epsilon$ and $k$, where $k$ is defined as $k = a(1- \epsilon^2)$.

Problem : The general equation for orbital motion is given by: $$x^2 + y^2 = k^2 – 2k\epsilon x + \epsilon^2 x^2$$ Where the $k$ is the same $k$ as in the last problem: $k = a(1-\epsilon^2) = \frac{L^2}{GMm^2}$. Show that when $\epsilon = 0$, this reduces to an equation for a circle. What is the radius of this circle?

Problem : Derive the formula for the area of an ellipse by integration.

Problem : Prove that for a point on an ellipse, the sum of the distances to each foci is a constant.

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