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

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

Identifying Conic Sections

Complete the squares to identify the conic sections in Exercises 69-76. Find their foci, vertices, centers, and asymptotes (as appropriate). If the curve is a parabola, find its directrix as well.

x² + y² + 4x + 2y = 1

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

Polar to Cartesian Equations

Sketch the lines in Exercises 23-28. Also, find a Cartesian equation for each line.

r cos (θ − 3π/4) = (√2)/2

Finding Parametric Equations and Tangent Lines

Find parametric equations for the given curve.

Line through (1,-2) with slope 3