90–94. Focal chords A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties.
The length of the latus rectum of a hyperbola centered at the origin is (2b²)/a = 2b√(1 - e²)
37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.
r cos θ = -4
39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin.
A hyperbola with vertices (±4, 0) and foci (±6, 0)
57–64. Graphing polar curves Graph the following equations. Use a graphing utility to check your work and produce a final graph.
r² = 4 sin θ
Air drop—inverse problem A plane traveling horizontally at 100 m/s over flat ground at an elevation of 4000 m must drop an emergency packet on a target on the ground. The trajectory of the packet is given by
x = 100t, y = −4.9t² + 4000, t ≥ 0
where the origin is the point on the ground directly beneath the plane at the moment of the release. How many horizontal meters before the target should the packet be released in order to hit the target?
25–30. Converting coordinates Express the following polar coordinates in Cartesian coordinates.
(4, 5π)
