16. Parametric Equations & Polar Coordinates

75–76. Graphs to polar equations Find a polar equation for each conic section. Assume one focus is at the origin.

Volume of a hyperbolic cap Consider the region R bounded by the right branch of the hyperbola x²/a² - y²/b² = 1 and the vertical line through the right focus.

a. What is the volume of the solid that is generated when R is revolved about the x-axis?

Volume of a hyperbolic cap Consider the region R bounded by the right branch of the hyperbola x²/a² - y²/b² = 1 and the vertical line through the right focus.

b. What is the volume of the solid that is generated when R is revolved about the y-axis?

90–94. Focal chords A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties.

The length of the latus rectum of a hyperbola centered at the origin is (2b²)/a = 2b√(1 - e²)

65–68. Eccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Graph the curve, labeling vertices, foci, asymptotes (if they exist), and directrices.

A hyperbola with vertices (0, ±2) and directrices y = ±1

53–57. Conic sections

b. Use analytical methods to determine the location of the foci, vertices, and directrices.

x² - y²/2 = 1

53–57. Conic sections

b. Use analytical methods to determine the location of the foci, vertices, and directrices.

4x² + 8y² = 16

53–57. Conic sections

d. Make an accurate graph of the curve.

x = 16y²

Parabola-hyperbola tangency: Let P be the parabola y = px² and H be the right half of the hyperbola x² - y² = 1.

a. For what value of p is P tangent to H?

90–94. Focal chords A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties.

Let L be the latus rectum of the parabola y ² =4px for p>0. Let F be the focus of the parabola, P be any point on the parabola to the left of L, and D be the (shortest) distance between P and L. Show that for all P, D+|FP|+ is a constant. Find the constant.

General equations for a circle Prove that the equations  

X = a cos t + b sin t,  y = c cos t + d sin t  

where a, b, c, and d are real numbers, describe a circle of radius R provided a² +c² =b² +d² = R² and ab+cd=0. 

61–64. Polar equations for conic sections Graph the following conic sections, labeling vertices, foci, directrices, and asymptotes (if they exist). Give the eccentricity of the curve. Use a graphing utility to check your work.

r = 3/(1 - 2 cos θ)

Theory and Examples

Tangents Find equations for the tangents to the circle (x − 2)² + (y − 1)² = 5 at the points where the circle crosses the coordinate axes.

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.

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)