16. Parametric Equations & Polar Coordinates

A polar conic section Consider the equation r² = sec2θ

a. Convert the equation to Cartesian coordinates and identify the curve.

53–57. Conic sections

a. Determine whether the following equations describe a parabola, an ellipse, or a hyperbola.

x²/4 + y²/25 = 1

Conic parameters: A hyperbola has eccentricity e = 2 and foci (0, ±2). Find the location of the vertices and directrices.

53–57. Conic sections

c. Find the eccentricity of the curve.

x²/4 + y²/25 = 1

53–57. Conic sections

c. Find the eccentricity of the curve.

y² - 4x² = 16

58–59. Tangent lines Find an equation of the line tangent to the following curves at the given point. Check your work with a graphing utility.

x²/16 - y²/9 = 1; (20/3, -4)

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

b. At what point does the tangency occur?

53–57. Conic sections

a. Determine whether the following equations describe a parabola, an ellipse, or a hyperbola.

x = 16y²

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