16. Parametric Equations & Polar Coordinates

How does the eccentricity determine the type of conic section?

Graphing Conic Sections

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

y² = −(8/3)x

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)

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)

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.

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.

53–57. Conic sections

d. Make an accurate graph of the curve.

x = 16y²

Hyperbolas and Eccentricity

In Exercises 17-24, find the eccentricity of the hyperbola. Then find and graph the hyperbola's foci and directrices.

y² − x² = 4

53–57. Conic sections

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

x = 16y²

Eccentricities and Directrices

Exercises 29–36 give the eccentricities of conic sections with one focus at the origin along with the directrix corresponding to that focus. Find a polar equation for each conic section.

e = 2, x = 4

If a parabola has the focus at $\(\left\)(2,4\(\right\))$ and a directrix line $x=-4$ , find the standard equation for the parabola.

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.

4x² - y² = 16

53–57. Conic sections

c. Find the eccentricity of the curve.

y² - 4x² = 16

How can you slice a vertically oriented 3D cone to get a 2D parabola?

Golden Gate Bridge Completed in 1937, San Francisco’s Golden Gate Bridge is 2.7 km long and weighs about 890,000 tons. The length of the span between the two central towers is 1280 m; the towers themselves extend 152 m above the roadway. The cables that support the deck of the bridge between the two towers hang in a parabola (see figure). Assuming the origin is midway between the towers on the deck of the bridge, find an equation that describes the cables. How long is a guy wire that hangs vertically from the cables to the roadway 500 m from the center of the bridge?