MATH 396 PROBLEMS 4

IGOR KRIZ

Regular problems:

1. Find the equation (in Cartesian coordiantes) of the ellipse with foci (c, 0), (−c, 0) and main axis a (i.e. the set of all points in the plane whose distances to the two foci add up to 2a).

2. The moon of a planet moves according the equation

r = \frac{1}{2 + cos(α)}

in polar coordinates with origin in the planet. What is the area swept by the moon (in the sense of Kepler’s law) in a quarter of a month (where the month is defined by that moon)? (Suppress units.) [Hint: if you choose to solve this problem by integrating in polar coordinates, remember the substitution t = tan(α/2). Alternately, you could use the major and minor axes.]

3. Consider the curve given by the equation r = α^{2} in polar coordinates. Calculate the area swept by the curve between the angles α = 0, α = π/3.

4. Consider the mapping f : ℝ^{2} → ℝ^{2} defined by

(u, v) ↦ (x, y)

where

x = u + v, y = v − u^{2}.

Let T be the triangle with vertices (0, 0), (2, 0), (0, 2). Let S = f[T] (the image of T under f). Using substitution, calculate

∫_{S}(x − y + 1)^{−2}dxdy.

Challenge problem:

5. Consider the parabola y = ax^{2} where a is a constant. The focus of the parabola is a point F on the y axis such that for any point P on the parabola, the angle between the line segment P F and the tangent at P is equal to the angle between the tangent at P and a vertical (=parallel to the y-axis) line through P.

(a) Find the focus of the parabola, and prove that it is a focus.

(b) Shifting the parabola so that the focus is in the origin, find its equation in polar coordinates.