16. Parametric Equations & Polar Coordinates

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)

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

A parabola symmetric about the y-axis that passes through the point (2, -6)

How does the eccentricity determine the type of conic section?

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 = 12y

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 (±2, 0) and asymptotes y = ±3x/2

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)

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

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

An ellipse with vertices (±5, 0), passing through the point (4, 3/5)

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

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. On every ellipse, there are exactly two points at which the curve has slope s, where s is any real number.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

d. The point on a parabola closest to the focus is the vertex.

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

69–72. Tangent lines Find an equation of the line tangent to the following curves at the given point.

x² = -6y; (-6, -6)

69–72. Tangent lines Find an equation of the line tangent to the following curves at the given point.

y² - x²/64 = 1; (6, -5/4)

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

e. The hyperbola y²/2 - x²/4 = 1 has no x-intercept.