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

Finding Parametric Equations and Tangent Lines

Find parametric equations for the given curve.

9x² + 4y² = 36

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

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

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)

Graphing Conic Sections

Find the eccentricities of the ellipses and hyperbolas in Exercises 59–62. Sketch each conic section. Include the foci, vertices, and asymptotes (as appropriate) in your sketch.

5y² − 4x² = 20