Reading: BEG 1.1.4, 1.1.5 (Dandelin spheres). We will study the results in 1.1.2 but at this time, we will use the approach in Ogilvy, 56-59, 73-85. On Friday, we will begin Chapter 2 of BEG with a return to isometries in the plane, but this time using coordinates more. For this, read BEG, Chapter 2, pp. 45-52.

1. (10 points) Let C be the conic in the (x, y) plane with eccentricity = 2 and with focus at (0, 0) and directrix line with equation x + y = 1. What is the equation of C? What is the other focus?
2. (10 points) Let C be the conic in the (x, y) plane with eccentricity = ½ and with focus at (0,0) and directrix line with equation x = 5. What is the equation of C? What is the center of point symmetry of C? If you shift the coordinates horizontally so that the center of symmetry is now at (0, 0), what is the new equation of C?
3. (10 points) Consider the constructions of the conics (the director circle construction) in GTC. Given points F1, F2 and P, how can you use this construction to construct the ellipse with Foci F1 and F2 which passes through P? How can you construct the hyperbola? Show that the tangent lines at P of the two conics are perpendicular. Hint: Use something about the optics and angle bisectors to build up the circle and the other stuff for the construction.
4. (5 points) BEG, page 52, problem 8.
5. (5 points) BEG, page 91, Ex. 3. Tell what kind of isometry each of the 2 transformations is and also what kind the composition is.
6. (5 points) BEG page 91, Ex 4. Tell what kind of isometry each of the 2 transformations are.
7. (5 points) BEG page 91, EX 5. Tell what kind of isometry each of the 2 transformations is and also what kind the composition is.