Chapter 7

Which one of the following is true? a. The parabola whose equation is \(x=2 y-y^{2}+5\) opens to the right. b. If the parabola whose equation is \(x=a y^{2}+b y+c\) has its vertex at \((3,2)\) and \(a>0,\) then it has no \(y\) -intercepts. c. Some parabolas that open to the right have equations that define \(y\) as a function of \(x .\) d. The graph of \(x=a(y-k)+h\) is a parabola with vertex at \((h, k)\)

Write an equation for the path of each of the following elliptical orbits. Then use a graphing utility to graph the two ellipses in the same viewing rectangle. Can you see why early astronomers had difficulty detecting that these orbits are ellipses rather than circles? Earth's orbit: \(\quad\) Length of major axis: 186 Length of minor axis: 185.8 million miles Mars's orbit: Length of major axis: 283.5 Length of minor axis: 278.5 million miles

Find the standard form of the equation of an ellipse with vertices at \((0,-6)\) and \((0,6),\) passing through \((2,-4)\)

Find the standard form of the equation of the hyperbola with vertices \((5,-6)\) and \((5,6),\) passing through \((0,9)\).

Write the standard form of the equation of a parabola whose points are equidistant from \(y=4\) and \((-1,0)\)

Find the equation of a hyperbola whose asymptotes are perpendicular.

Consult the research department of your library or the Internet to find an example of architecture that incorporates one or more conic sections in its design. Share this example with other group members. Explain precisely how conic sections are used. Do conic sections enhance the appeal of the architecture? In what ways?