Do all problems. Each problem is worth 25 points.

If you can, write down the entries of matrices as exact real numbers using geometry. Writing a matrix with trig functions will be correct, but somewhat less valuable.

1. Let F(x) = Mx + b be the rotation of x by 120 degrees with center (0,0). Write down the matrix for M and the vector b.
2. Let G(x) = Nx + c be the reflection of x in the line y = 2 Write down the matrix for N and the vector c.
3. Write the matrix formula for FG(x). [FG means F composed with G.]

Answer these questions using the figure "Barycentric Coordinate Figure." The barycentric coordinates used in this problem are with respect to the triangle ABC.

1. (No computation needed). Circle and label the point P with barycentric coordinates (1/2,1/3,1/6) and Q with barycentric coordinates (-1/3,1,1/3).
2. (Computation needed). Let Z be the intersection of lines BY and CX. What are the barycentric coordinates of Z?

1. State the definition of a conic using the focus-directrix-eccentricity definition. Also state which eccentricity defines ellipses, which defines parabolas, and which defines hyperbolas.
2. M has columns M1, M2, where M1 = (1/2,1/3) and M2 = (1/3, -1/2). [Should be written as column vectors.] Is this the matrix of a reflection? Give quick reason.
3. N has columns N1 and N2, where N1 = (3/5,4/5) and N2 = (4/5,3/5). [Should be written as column vectors.] Is this the matrix of a reflection? Give quick reason

1. State the definition of an ellipse in terms of two foci F1 and F2 and the distances from the foci to the points P of the ellipse.
2. Construct using the circle construction a "random" point P of the ellipse with the two foci F1, F2 and the major axis d shown in the figure. Don't construct P on one of the axes, but rather construct P using a circle so that if a point if animated along the circle, then P would trace out the ellipse.
3. Indicate briefly why the point P satisfies the definition of an ellipse.