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

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

Lines

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

r cos (θ + π/3) = 2

Identifying Parametric Equations in the Plane

Exercises 1–6 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 and indicate the direction of motion and the portion traced by the particle.

x = 4 cos t, y = 9 sin t, 0 ≤ t ≤ 2π

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.

Line through (1,-2) with slope 3