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

Finding Parametric Equations and Tangent Lines

Find parametric equations for the given curve.

9x² + 4y² = 36

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 = √t, y = 1 − √t, t ≥ 0

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

Area in Polar Coordinates

Find the areas of the regions in the polar coordinate plane described in Exercises 47–50.

Inside the cardioid r = 2(1 + sin θ) and outside the circle r = 2 sin θ

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