Finding Cartesian from Parametric Equations


In Exercises 19–24, match the parametric equations with the parametric curves labeled A through F.


x = cos t, y = sin 3t

Centroids

Find the coordinates of the centroid of the curve x = cos t, y = t + sin t, 0 ≤ t ≤ π.

Finding Lengths of Polar Curves

Find the lengths of the curves in Exercises 21–28.

The curve r = cos³(θ/3), 0 ≤ θ ≤ π/4

Identifying Graphs

Match the parabolas in Exercises 1−4 with the following equations: x² = 2y, x² = −6y, y² = 8x, y² = −4x

Then find each parabola's focus and directrix.

Ellipses and Eccentricity

Exercises 9–12 give the foci or vertices and the eccentricities of ellipses centered at the origin of the xy-plane. In each case, find the ellipse’s standard-form equation in Cartesian coordinates.

Vertices: (±10,0)

Eccentricity: 0.24

Tangent Lines to Parametrized Curves

In Exercises 1−14, find an equation for the line tangent to the curve at the point defined by the given value of t. Also, find the value of d²y/dx² at this point.

x = sec² t − 1, y = tan t, t = −π/4

Polar to Cartesian Equations

Replace the polar equations in Exercises 27–52 with equivalent Cartesian equations. Then describe or identify the graph.

r = 3 cos θ