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

Finding Polar Areas

Find the areas of the regions in Exercises 9–18.

Shared by the circles r = 1 and r = 2 sin θ

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

Lines

Sketch the lines in Exercises 45–48 and find Cartesian equations for them.

r cos (θ + π/3) = 2

Hyperbolas and Eccentricity

Exercises 25–28 give the eccentricities and the vertices or foci of hyperbolas centered at the origin of the xy-plane. In each case, find the hyperbola’s standard-form equation in Cartesian coordinates.

Eccentricity: 1.25

Foci: (0, ±5)

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