Chapter 10

SATELLITES For Exercises \(33-35,\) use the following information. Two satellites are placed in orbit about Earth. The equations of the two orbits \(\operatorname{are} \frac{x^{2}}{(300)^{2}}+\frac{y^{2}}{(900)^{2}}=1\) and \(\frac{x^{2}}{(600)^{2}}+\frac{y^{2}}{(690)^{2}}=1,\) where distances are in kilometers and Earth is the center of each curve. Use a graphing calculator to estimate the intersection points of the two orbits.

Write each equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola. Then graph the equation. $$ x^{2}-8 y+y^{2}+11=0 $$

Identify the coordinates of the vertex and focus, the equations of the axis of symmetry and directrix, and the direction of opening of the parabola with the given equation. Then find the length of the latus rectum and graph the parabola. $$ y=3 x^{2}-24 x+50 $$

Write an equation for the circle that satisfies each set of conditions. center at \((-8,-7),\) tangent to \(y\) -axis

Write an equation for an ellipse with its center at (2, -5) and a horizontal major axis.

For Exercises \(34-37,\) use the following information. A hyperbola with asymptotes that are not perpendicular is called a nonrectangular hyperbola. Most of the hyperbolas you have studied so far are nonrectangular. A rectangular hyperbola is a hyperbola with perpendicular asymptotes. For example, the graph of \(x^{2}-y^{2}=1\) is a rectangular hyperbola. The graphs of equations of the form \(x y=c,\) where \(c\) is a constant, are rectangular hyperbolas with the coordinate axes as their asymptotes. Find the coordinates of the vertices of the graph of \(x y=2\)

SATELLITES For Exercises \(33-35,\) use the following information. Two satellites are placed in orbit about Earth. The equations of the two orbits \(\operatorname{are} \frac{x^{2}}{(300)^{2}}+\frac{y^{2}}{(900)^{2}}=1\) and \(\frac{x^{2}}{(600)^{2}}+\frac{y^{2}}{(690)^{2}}=1,\) where distances are in kilometers and Earth is the center of each curve. Compare the orbits of the two satellites.

Write each equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola. Then graph the equation. $$ 9 y^{2}+18 y=25 x^{2}+216 $$

Identify the coordinates of the vertex and focus, the equations of the axis of symmetry and directrix, and the direction of opening of the parabola with the given equation. Then find the length of the latus rectum and graph the parabola. $$ y=-2 x^{2}+5 x-10 $$

Write an equation for the circle that satisfies each set of conditions. center at \((4,2),\) tangent to \(x\) -axis

Find an equation for the ellipse with foci at \((\sqrt{3}, 0)\) and \((-\sqrt{3}, 0)\) that passes through \((0,3)\)

For Exercises \(34-37,\) use the following information. A hyperbola with asymptotes that are not perpendicular is called a nonrectangular hyperbola. Most of the hyperbolas you have studied so far are nonrectangular. A rectangular hyperbola is a hyperbola with perpendicular asymptotes. For example, the graph of \(x^{2}-y^{2}=1\) is a rectangular hyperbola. The graphs of equations of the form \(x y=c,\) where \(c\) is a constant, are rectangular hyperbolas with the coordinate axes as their asymptotes. Graph \(x y=-2\)

Write each equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola. Then graph the equation. $$ 3 x^{2}+4 y^{2}+8 y=8 $$

Identify the coordinates of the vertex and focus, the equations of the axis of symmetry and directrix, and the direction of opening of the parabola with the given equation. Then find the length of the latus rectum and graph the parabola. $$ x=-4 y^{2}+6 y+2 $$

Write an equation for the circle that satisfies each set of conditions. center in the first quadrant; tangent to \(x=-3, x=5,\) and the \(x\) -axis

OPEN ENDED Find two points that are \(\sqrt{29}\) units apart.

For Exercises \(34-37,\) use the following information. A hyperbola with asymptotes that are not perpendicular is called a nonrectangular hyperbola. Most of the hyperbolas you have studied so far are nonrectangular. A rectangular hyperbola is a hyperbola with perpendicular asymptotes. For example, the graph of \(x^{2}-y^{2}=1\) is a rectangular hyperbola. The graphs of equations of the form \(x y=c,\) where \(c\) is a constant, are rectangular hyperbolas with the coordinate axes as their asymptotes. Describe the transformations that can be applied to the graph of \(x y=2\) to obtain the graph of \(x y=-2\) .

Write each equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola. Then graph the equation. $$ x^{2}+4 y^{2}-11=2(4 y-x) $$

Identify the coordinates of the vertex and focus, the equations of the axis of symmetry and directrix, and the direction of opening of the parabola with the given equation. Then find the length of the latus rectum and graph the parabola. $$ x=5 y^{2}-10 y+9 $$

Write an equation for the circle that satisfies each set of conditions. center in the second quadrant; tangent to \(y=-1, y=9,\) and the \(y\) -axis

Winona is making an elliptical target for throwing darts. She wants the target to be 27 inches wide and 15 inches high. Which equation should Winona use to draw the target? A. \(\frac{y^{2}}{13.5}+\frac{x^{2}}{7.5}=1\) B. \(\frac{y^{2}}{182.25}+\frac{x^{2}}{56.25}=1\) C. \(\frac{y^{2}}{56.25}+\frac{x^{2}}{182.25}=1\) D. \(\frac{y^{2}}{7.5}+\frac{x^{2}}{13.5}=1\)

OPEN ENDED Find and graph a counterexample to the following statement. If the equation of a hyperbola is \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1,\) then \(a^{2} \geq b^{2} .\)

Write each equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola. Then graph the equation. $$ y+x^{2}=-(8 x+23) $$

Identify the coordinates of the vertex and focus, the equations of the axis of symmetry and directrix, and the direction of opening of the parabola with the given equation. Then find the length of the latus rectum and graph the parabola. $$ y=\frac{1}{2} x^{2}-3 x+\frac{19}{2} $$

The Rose Bowl is located about 35 miles west and about 40 miles north of downtown Los Angeles. Suppose an earthquake occurs with its epicenter about 55 miles from the stadium. Assume that the origin of a coordinate plane is located at the center of downtown Los Angeles. Write an equation for the set of points that could be the epicenter of the earthquake.

What is the standard form of the equation of the conic given below? \(2 x^{2}-4 y^{2}-8 x-24 y-16=0\) F. \(\frac{(x-4)^{2}}{11}-\frac{(y+3)^{2}}{3}=1\) G. \(\frac{(y-3)^{2}}{3}-\frac{(x-2)^{2}}{6}=1\) H. \(\frac{(y+3)^{2}}{4}-\frac{(x+2)^{2}}{5}=1\) J. \(\frac{(x-4)^{2}}{11}+\frac{(y+3)^{2}}{3}=1\)

Write an equation for the circle that satisfies each set of conditions. center \((3,-2),\) radius 5 units

Describe how the graph of \(y^{2}-\frac{x^{2}}{k^{2}}=1\) changes as \(|k|\) increases.

Write a system of equations that satisfies each condition. Use a graphing calculator to verify that you are correct. a circle and an ellipse that intersect in four points

Write each equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola. Then graph the equation. $$ 6 x^{2}-24 x-5 y^{2}-10 y-11=0 $$

Identify the coordinates of the vertex and focus, the equations of the axis of symmetry and directrix, and the direction of opening of the parabola with the given equation. Then find the length of the latus rectum and graph the parabola. $$ x=-\frac{1}{3} y^{2}-12 y+15 $$

Write an equation for the circle that satisfies each set of conditions. endpoints of a diameter at (5, -9) and (3, 11)

CHALLENGE A hyperbola with a horizontal transverse axis contains the point at \((4,3) .\) The equations of the asymptotes are \(y-x=1\) and \(y+x=5\) Write the equation for the hyperbola.

Write each equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola. Then graph the equation. $$ 25 y^{2}+9 x^{2}-50 y-54 x=119 $$

UMBRELLAS A beach umbrella has an arch in the shape of a parabola that opens downward. The umbrella spans 9 feet across and 1\(\frac{1}{2}\) feet high. Write an equation of a parabola to model the arch, assuming that the origin is at the point where the pole and umbrella meet, beneath the vertex of the arch.

For Exercises \(40-43,\) use the following information. since a circle is not the graph of a function, you cannot enter its equation directly into a graphing calculator. Instead, you must solve the equation for \(y .\) The result will contain a \pm symbol, so you will have two functions. Solve \((x+3)^{2}+y^{2}=16\) for \(y\)

ACT/SAT Point \(D(5,-1)\) is the midpoint of segment \(C E .\) If point \(C\) has coordinates \((3,2),\) what are the coordinates of point \(E ?\) $$ \begin{array}{l}{\mathbf{A}(8,1)} \\ {\mathbf{B}(7,-4)} \\\ {\mathbf{C}(2,-3)} \\ {\mathbf{D}\left(4, \frac{1}{2}\right)}\end{array} $$

Write an equation of a parabola with vertex \((3,1)\) and focus \(\left(3,1 \frac{1}{2}\right) .\) Then draw the graph.

Explain how hyperbolas and parabolas are different. Include differences in the graphs of hyperbolas and parabolas and differences in the reflective properties of hyperbolas and parabolas.

Without writing the equation in standard form, state whether the graph of each equation is a parabola, circle, ellipse, or hyperbola. $$ 5 x^{2}+6 x-4 y=x^{2}-y^{2}-2 x $$

REASONING How do you change the equation of the parent function \(y=x^{2}\) to shift the graph to the right?

REVIEW If \(\log _{10} x=-3,\) what is the value of \(x ?\) $$ \begin{array}{ll}{\mathbf{F} \quad x=1000} & {\mathbf{H} x=\sqrt{\frac{1}{100}}} \\ {\mathbf{G} x=\frac{1}{1000}} & {\mathbf{J} \quad x=-\sqrt{\frac{1}{100}}}\end{array} $$

REASONING Sketch a parabola and an ellipse that intersect in exactly three points.

ACT/SAT The foci of the graph are at \((\sqrt{13}, 0)\) and \((-\sqrt{13}, 0) .\) Which equation does the graph represent? $$ \begin{array}{l}{\text { A } \frac{x^{2}}{9}-\frac{y^{2}}{4}=1} \\ {\text { B } \frac{x^{2}}{3}-\frac{y^{2}}{2}=1} \\ {\text { C } \frac{x^{2}}{3}-\frac{y^{2}}{\sqrt{13}}=1} \\ {\text { D } \frac{x^{2}}{9}-\frac{y^{2}}{13}=1}\end{array} $$

Without writing the equation in standard form, state whether the graph of each equation is a parabola, circle, ellipse, or hyperbola. $$ 2 x^{2}+12 x+18-y^{2}=3\left(2-y^{2}\right)+4 y $$

OPEN ENDED Write an equation for a parabola that opens to the left. Use the parent graph to sketch the graph of your equation.

For Exercises \(40-43,\) use the following information. since a circle is not the graph of a function, you cannot enter its equation directly into a graphing calculator. Instead, you must solve the equation for \(y .\) The result will contain a \pm symbol, so you will have two functions. Graph \((x+3)^{2}+y^{2}=16\) on a graphing calculator.

COMPUTERS Suppose a computer that costs \(\$ 3000\) new is only worth \(\$ 600\) after 3 years. What is the average annual rate of depreciation?

OPEN ENDED Write a system of quadratic equations for which \((2,6)\) is a solution.

REVIEW To begin a game, Nate must randomly draw a red, blue, green, or yellow game piece, and a tile from a group of 26 tiles labeled with all the letters of the alphabet. What is the probability that Nate will draw the green game piece and a tile with a letter from his name? $$ \begin{array}{ll}{\mathbf{F} \frac{1}{26}} & {\mathbf{H} \frac{3}{52}} \\\ {\mathbf{G} \frac{1}{13}} & {\mathbf{J} \frac{1}{2}}\end{array} $$