1. Ellipses

(a) State the definition of an ellipse (the one in terms of distances in the plane and two focus points, not the intersection of a cone with a plane).

(b) Draw a figure consisting of 2 points F and G and a circle centered at G with F inside the circle. Construct a "random" point P on the ellipse with foci F and G with distance sum from the definition = the radius of the circle. Also construct the line p through P which is tangent to the ellipse. Write down the major steps of your construction so that it is clear what you did and how other points would be constructed.

(c) Tell why the construction works. Specifically, why is the point P on the ellipse and why does p only intersect the ellipse at P?

(d) State the optical property of the ellipse. What beams of light will focus at F?

2. Tetrahedra in spheres

(a) What is the Euclidean (3D) distance between two vertices of the tetrahedron? (Hint: the 4 vertices of a regular tetrahedron also are 4 of the 8 vertices of a cube, so you can work with a cube.)

(b) What is the spherical distance, measured in degrees or radians, between these two vertices? (You can leave this answer in terms of an inverse trig function.)