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
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
Lengths of Curves
Find the lengths of the curves in Exercises 13–19.
x = 5 cos t − cos 5t, y = 5 sin t − sin 5t, 0 ≤ t ≤ π/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
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
Finding Parametric Equations and Tangent Lines
Find parametric equations for the given curve.
Line through (1,-2) with slope 3
