Chapter 7

In Exercises 43-50, convert each equation to standard form by completing the square on \(x\) and \(y .\) Then graph the hyperbola. Locate the foci and find the equations of the asymptotes. $$x^{2}-y^{2}-2 x-4 y-4=0$$

Convert each equation to standard form by completing the square on \(x\) or \(y .\) Then find the vertex, focus, and directrix of the parabola. Finally, graph the parabola. $$ x^{2}-2 x-4 y+9=0 $$

Graph each ellipse and give the location of its foci. $$\frac{(x-4)^{2}}{4}+\frac{y^{2}}{25}=1$$

Convert each equation to standard form by completing the square on \(x\) and \(y .\) Then graph the hyperbola. Locate the foci and find the equations of the asymptotes. $$4 x^{2}-y^{2}+32 x+6 y+39=0$$

Graph each ellipse and give the location of its foci. $$\frac{(x+3)^{2}}{9}+(y-2)^{2}=1$$

Convert each equation to standard form by completing the square on \(x\) and \(y .\) Then graph the hyperbola. Locate the foci and find the equations of the asymptotes. $$16 x^{2}-y^{2}+64 x-2 y+67=0$$

Graph each ellipse and give the location of its foci. $$\frac{(x+2)^{2}}{16}+(y-3)^{2}=1$$

Convert each equation to standard form by completing the square on \(x\) and \(y .\) Then graph the hyperbola. Locate the foci and find the equations of the asymptotes. $$9 y^{2}-4 x^{2}-18 y+24 x-63=0$$

Graph each ellipse and give the location of its foci. $$\frac{(x-1)^{2}}{2}+\frac{(y+3)^{2}}{5}=1$$

Convert each equation to standard form by completing the square on \(x\) and \(y .\) Then graph the hyperbola. Locate the foci and find the equations of the asymptotes. $$4 x^{2}-9 y^{2}-16 x+54 y-101=0$$

Graph each ellipse and give the location of its foci. $$\frac{(x+1)^{2}}{2}+\frac{(y-3)^{2}}{5}=1$$

Convert each equation to standard form by completing the square on \(x\) and \(y .\) Then graph the hyperbola. Locate the foci and find the equations of the asymptotes. $$4 x^{2}-9 y^{2}+8 x-18 y-6=0$$

Graph each ellipse and give the location of its foci. $$9(x-1)^{2}+4(y+3)^{2}=36$$

Convert each equation to standard form by completing the square on \(x\) and \(y .\) Then graph the hyperbola. Locate the foci and find the equations of the asymptotes. $$4 x^{2}-25 y^{2}-32 x+164=0$$

The reflector of a flashlight is in the shape of a parabolic Surface. The casting has a diameter of 4 inches and a depth of 1 inch. How far from the vertex should the light bulb be placed?

Graph each ellipse and give the location of its foci. $$36(x+4)^{2}+(y+3)^{2}=36$$

Convert each equation to standard form by completing the square on \(x\) and \(y .\) Then graph the hyperbola. Locate the foci and find the equations of the asymptotes. $$9 x^{2}-16 y^{2}-36 x-64 y+116=0$$

The reflector of a flashlight is in the shape of a parabolic surface. The casting has a diameter of 8 inches and a depth of 1 inch. How far from the vertex should the light bulb be placed?

Convert each equation to standard form by completing the square on \(x\) and \(y .\) Then graph the ellipse and give the location of its foci. $$9 x^{2}+25 y^{2}-36 x+50 y-164=0$$

An explosion is recorded by two microphones that are 1 mile apart. Microphone \(M_{1}\) received the sound 2 seconds before microphone \(M_{2} .\) Assuming sound travels at 1100 feet per second, determine the possible locations of the explosion relative to the location of the microphones.

A satellite dish, like the one shown at the top of the next column, is in the shape of a parabolic surface. Signals coming from a satellite strike the surface of the dish and are reflected to the focus, where the receiver is located. The satellite dish shown has a diameter of 12 feet and a depth of 2 feet. How far from the base of the dish should the receiver be placed?

Radio towers \(A\) and \(B, 200\) kilometers apart, are situated along the coast, with \(A\) located due west of \(B\). Simultaneous radio signals are sent from each tower to a ship, with the signal from \(B\) received 500 microseconds before the signal from \(A\). a. Assuming that the radio signals travel 300 meters per microsecond, determine the equation of the hyperbola on which the ship is located. b. If the ship lies due north of tower \(B,\) how far out at sea is it?

Convert each equation to standard form by completing the square on \(x\) and \(y .\) Then graph the ellipse and give the location of its foci. $$9 x^{2}+16 y^{2}-18 x+64 y-71=0$$

An architect designs two houses that are shaped and positioned like a part of the branches of the hyperbola whose equation is \(625 y^{2}-400 x^{2}=250,000,\) where \(x\) and \(y\) are in yards. How far apart are the houses at their closest point?

Convert each equation to standard form by completing the square on \(x\) and \(y .\) Then graph the ellipse and give the location of its foci. $$x^{2}+4 y^{2}+10 x-8 y+13=0$$

Convert each equation to standard form by completing the square on \(x\) and \(y .\) Then graph the ellipse and give the location of its foci. $$4 x^{2}+y^{2}+16 x-6 y-39=0$$

What is a hyperbola?

Convert each equation to standard form by completing the square on \(x\) and \(y .\) Then graph the ellipse and give the location of its foci. $$4 x^{2}+25 y^{2}-24 x+100 y+36=0$$

Describe how to graph \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\)

Describe how to locate the foci of the graph of \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\)

What is a parabola?

Describe one similarity and one difference between the graphs of \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\) and \(\frac{y^{2}}{9}-\frac{x^{2}}{1}=1\)

Explain how to use \(y^{2}=8 x\) to find the parabola's focus and directrix.

Describe one similarity and one difference between the \(\operatorname{graphs~of~} \frac{x^{2}}{9}-\frac{y^{2}}{1}=1\) and \(\frac{(x-3)^{2}}{9}-\frac{(y+3)^{2}}{1}=1\).

If you are given the standard form of the equation of a parabola with vertex at the origin, explain how to determine if the parabola opens to the right, left, upward, or downward.

How can you distinguish an ellipse from a hyperbola by looking at their equations?

Describe one similarity and one difference between the graphs of \(y^{2}=4 x\) and \((y-1)^{2}=4(x-1)\)

What is an ellipse?

In \(1992,\) a NASA team began a project called Spaceguard Survey, calling for an international watch for comets that might collide with Earth. Why is it more difficult to detect a possible "doomsday comet" with a hyperbolic orbit than one with an elliptical orbit?

How can you distinguish parabolas from other conic sections by looking at their equations?

Describe how to graph \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\).

Describe how to locate the foci for \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\).

Describe one similarity and one difference between the graphs of \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\) and \(\frac{(x-1)^{2}}{25}+\frac{(y-1)^{2}}{16}=1\).

Use a graphing utility to graph \(\frac{x^{2}}{4}-\frac{y^{2}}{9}=0 .\) Is the graph a hyperbola? In general, what is the graph of \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=0 ?\)

Use a graphing utility to graph the parabolas.Write the given equation as a quadratic equation in \(y\) and use the quadratic formula to solve for \(y .\) Enter each of the equations to produce the complete graph. $$y^{2}+2 y-6 x+13=0$$

Graph \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) and \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=-1\) in the same viewing rectangle for values of \(a^{2}\) and \(b^{2}\) of your choice. Describe the relationship between the two graphs.

Use a graphing utility to graph the parabolas.Write the given equation as a quadratic equation in \(y\) and use the quadratic formula to solve for \(y .\) Enter each of the equations to produce the complete graph. $$y^{2}+10 y-x+25=0$$

Write \(\quad 4 x^{2}-6 x y+2 y^{2}-3 x+10 y-6=0\) as a quadratic equation in \(y\) and then use the quadratic formula to express \(y\) in terms of \(x .\) Graph the resulting two equations using a graphing utility in a \([-50,70,10]\) by \([-30,50,10]\) viewing rectangle. What effect does the \(x y\) -term have on the graph of the resulting hyperbola? What problems would you encounter if you attempted to write the given equation in standard form by completing the square?

Graph \(\frac{x^{2}}{16}-\frac{y^{2}}{9}=1\) and \(\frac{x|x|}{16}-\frac{y|y|}{9}=1\) in the same viewing rectangle. Explain why the graphs are not the same.

Which one of the following is true? a. If one branch of a hyperbola is removed from a graph, then the branch that remains must define \(y\) as a function of \(x .\) b. All points on the asymptotes of a hyperbola also satisfy the hyperbola's equation. c. The graph of \(\frac{x^{2}}{9}-\frac{y^{2}}{4}=1\) does not intersect the line \(y=-\frac{2}{3} x\) d. Two different hyperbolas can never share the same asymptotes.