Shifting Conic Sections


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


The hyperbola (y²/4) − (x²/5) = 1 is shifted 2 units down to generate the hyperbola (y + 2)²/4 − x²/5 = 1.
a. Find the center, foci, vertices, and asymptotes of the new hyperbola.

Polar Coordinates

Plot the following points, given in polar coordinates. Then find all the polar coordinates of each point.

a. (2, π/2)

Graphing Conic Sections

Sketch the parabolas in Exercises 55–58. Include the focus and directrix in each sketch.

y² = −(8/3)x

Finding Parametric Equations

Find parametric equations and a parameter interval for the motion of a particle that starts at (a, 0) and traces the circle x²+y²=a².

a. once clockwise.

(There are many ways to do these, so your answers may not be the same as the ones at the back of the text.)

Theory and Examples

Volume Find the volume of the solid generated by revolving the region enclosed by the ellipse 9x² + 4y² = 36 about the y−axis.

Cartesian to Polar Coordinates

Find the polar coordinates, 0 ≤ θ < 2π and r ≥ 0, of the following points given in Cartesian coordinates.

b. (-3,0)

Cycloid

a. Find the length of one arch of the cycloid x = a(t − sin t), y = a(1 − cos t).