Name and Student ID:
Homework 3, Analytic Geometry and Matrices
Problems concerning coordinates:
1. Consider cylindrical coordinate Φ : rθu → xyz. Sketch the pre-image, in rθu space, of the region bounded by x2+ y2 ≤ 1, y ≥ 0, y ≤ x, and 0 ≤ z ≤ 1 in xyz space. (without concern on the boundary requirement for one-to-one property of Φ):
2. Consider spherical coordinate Φ : ρθφ → xyz. Sketch the pre-image, in ρθφ space, of the lower half of the ball x2+ y2+ z2 ≤ 1 in xyz space. (without concern on the boundary requirement for one-to-one property of Φ):
Problems concerning conic sections:
1. Given φ ∈ R, prove, using polar coordinates, that the map
R(x, y) = (x cos φ − y sin φ , x sin φ + y cos φ)
rotates every point on R2 by an angle φ, counterclockwise. What is the inverse of R?
2. Write down a rotation R : R2 → R2 that takes two opposite points ±(α, β) to two opposite points on ±(c, 0) on x-axis. What is c?
3. Using previous two problems, write down the equation of the ellipse with foci ±(α, β) and length of major axis 2a for some a ≥pα2+ β2.
4. Write down the equation of an ellipse with general foci (p, q), and (r, s) and length of major axis 2a for some a ≥ 12
q
(p − r)2+ (q − s)2. You do NOT need to expand nor simplify your final quadratic polynomial.
5. Repeat problem 4 for hyperbola.
6. (Extra Credit - 15 points) Write down the equation for parabola with focus (p, q) and directrix y = ax+b. Again, you do NOT need to expand nor simplify your final quadratic polynomial.
