16. Parametric Equations & Polar Coordinates

57–62. Polar equations for conic sections Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work.

r = 3/(2 + cos θ)

Sketch a graph of the circle based on the following equation: $x^2+\(\left\)(y-1\(\right\))^2=9$

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?

How can you slice a vertically oriented 3D cone with a 2D plane to get a circle?

13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.

x²/9 + y²/4 = 1

57–62. Polar equations for conic sections Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work.

r = 1/(2 - 2 sin θ)

Determine the vertices and foci of the following ellipse: $\(\frac{x^2}{9}\)+\(\frac{y^2}{16}\)=1$.

Given the ellipse equation $\(\frac{x^2}{16}\)+\(\frac{y^2}{4}\)=1$, determine the magnitude of the semi-major axis (a) and the semi-minor axis (b).

39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin. 

Tangent lines for a hyperbola Find an equation of the line tangent to the hyperbola x²/a² + y²/b² = 1 at the point (x₀, y₀)

Graph the parabola $8\(\left\)(x+1\(\right\))=\(\left\)(y-2\(\right\))^2$ , and find the focus point and directrix line.

31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.

A parabola with focus at (3, 0)

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)

31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.

A parabola that opens to the right with directrix x = -4

39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin. 

A hyperbola with vertices (±4, 0) and foci (±6, 0)