16. Parametric Equations & Polar Coordinates

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? 

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)

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?

Determine the vertices and foci of the following ellipse: $\(\frac{x^2}{49}\)+\(\frac{y^2}{36}\)=1$.

53–57. Conic sections

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

x² - y²/2 = 1

Find the equation for the following circle:

Graphing Conic Sections

Find the eccentricities of the ellipses and hyperbolas in Exercises 59–62. Sketch each conic section. Include the foci, vertices, and asymptotes (as appropriate) in your sketch.

5y² − 4x² = 20

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.

25y² - 4x² = 100

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

53–57. Conic sections

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

4x² + 8y² = 16

Determine the vertices and foci of the ellipse $\(\left\)(x+1\(\right\))^2+\(\frac{\left(y-2\right)^2}{4}\)=1$.

Find the equation for a hyperbola with a center at $\(\left\)(0,0\(\right\))$, focus at $\(\left\)(0,-6\(\right\))$ and vertex at $\(\left\)(0,4\(\right\))$ .

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

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)