Ellipses


Exercises 25 and 26 give information about the foci and vertices of ellipses centered at the origin of the xy−plane. In each case, find the ellipse's standard−form equation from the given information.


Foci: ( ±√2, 0) Vertices: (±2,0)

Shifting Conic Sections

Find the center, foci, vertices, asymptotes, and radius, as appropriate, of the conic sections in Exercises 57-68.

9x² + 6y² + 36y = 0

Finding Cartesian from Parametric Equations

Exercises 1–18 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. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

x = 2 sinh t, y = 2 cosh t, −∞<t<∞

Circles

Sketch the circles in Exercises 53–56. Give polar coordinates for their centers and identify their radii.

r = −2 cos θ

Parabolas

Exercises 9-16 give equations of parabolas. Find each parabola's focus and directrix. Then sketch the parabola. Include the focus and directrix in your sketch.

x = −3y²

Parabolas

Exercises 9-16 give equations of parabolas. Find each parabola's focus and directrix. Then sketch the parabola. Include the focus and directrix in your sketch.

x² = 6y

Shifting Conic Sections

You may wish to review Section 1.2 before solving Exercises 39-56.

Exercises 53-56 give equations for hyperbolas and tell how many units up or down and to the right or left each hyperbola is to be shifted. Find an equation for the new hyperbola, and find the new center, foci, vertices, and asymptotes.

x²/4 − y²/5 = 1, right 2, up 2