Graphing Conic Sections
Exercises 63-68 give equations for conic sections and tell how many units up or down and to the right or left each curve is to be shifted. Find an equation for the new conic section, and find the new foci, vertices, centers, and asymptotes, as appropriate. If the curve is a parabola, find the new directrix as well.
x²/169 + y²/144 = 1, right 5, up 12
Cartesian to Polar Equations
Replace the Cartesian equations in Exercises 53–66 with equivalent polar equations.
(x + 2)² + (y − 5)² = 16"
Finding Cartesian from Parametric Equations
Exercises 1–18 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.
x=√(t+1), y=√t, t ≥ 0
Hyperbolas and Eccentricity
In Exercises 17-24, find the eccentricity of the hyperbola. Then find and graph the hyperbola's foci and directrices.
y² − x² = 4
Lines
Sketch the lines in Exercises 45–48 and find Cartesian equations for them.
r cos (θ + π/3) = 2
Polar Coordinates
Exercises 19–22 give the eccentricities of conic sections with one focus at the origin of the polar coordinate plane, along with the directrix for that focus. Find a polar equation for each conic section.
e = 1/3, r sin θ = −6
