Chapter 10

The rounded top of the window is the top half of an ellipse. Write an equation for the ellipse if the origin is at the midpoint of the bottom edge of the window. \(16 x^{2}+25 y^{2}+32 x-150 y=159\)

Find the midpoint of the line segment with endpoints at the given coordinates. Then find the distance between the points. $$ \left(0, \frac{1}{5}\right),\left(\frac{3}{5},-\frac{3}{5}\right) $$

Graph each system of equations. Use the graph to solve the system. $$ \begin{array}{ll}{\text { a. } 4 x-3 y=0} & {\text { b. } y=5-x^{2}} \\\ {x^{2}+y^{2}=25} & {y=2 x^{2}+2}\end{array} $$

Find the coordinates of the vertices and foci and the equations of the asymptotes for the hyperbola with the given equation. Then graph the hyperbola. $$ \frac{(x+1)^{2}}{4}-\frac{(y+3)^{2}}{9}=1 $$

For Exercises \(24-27,\) use the equation \(x=3 y^{2}+4 y+1\) What are the coordinates of the vertex?

Find the center and radius of the circle with the given equation. Then graph the circle. $$ x^{2}+y^{2}+6 y=-50-14 x $$

Write an equation for the ellipse that satisfies each set of conditions. major axis 16 units long and parallel to \(x\)-axis, minor axis 9 units long, center at (5, 4)

Find the midpoint of the line segment with endpoints at the given coordinates. Then find the distance between the points. $$ (2 \sqrt{3},-5),(-3 \sqrt{3}, 9) $$

ASTRONOMY For Exercises 27 and 28 , use the following information. The orbit of Pluto can be modeled by the equation \(\frac{x^{2}}{39.5^{2}}+\frac{y^{2}}{38.3^{2}}=1,\) where the units are astronomical units. Suppose a comet is following a path modeled by the equation \(x=y^{2}+20\) Find the point(s) of intersection of the orbits of Pluto and the comet. Round to the nearest tenth.

Find the coordinates of the vertices and foci and the equations of the asymptotes for the hyperbola with the given equation. Then graph the hyperbola. $$ \frac{(x+6)^{2}}{36}-\frac{(y+3)^{2}}{9}=1 $$

For Exercises \(24-27,\) use the equation \(x=3 y^{2}+4 y+1\) How does the graph compare to the graph of the parent function \(x=y^{2} ?\)

Find the center and radius of the circle with the given equation. Then graph the circle. $$ x^{2}+y^{2}-6 y-16=0 $$

Write an equation for the ellipse that satisfies each set of conditions. endpoints of major axis at (10, 2) and (-8, 2), foci at (6, 2) and (-4, 2)

Find the midpoint of the line segment with endpoints at the given coordinates. Then find the distance between the points. $$ \left(\frac{2 \sqrt{3}}{3}, \frac{\sqrt{5}}{4}\right),\left(-\frac{2 \sqrt{3}}{3}, \frac{\sqrt{5}}{2}\right) $$

The orbit of Pluto can be modeled by the equation \(\frac{x^{2}}{39.5^{2}}+\frac{y^{2}}{38.3^{2}}=1,\) where the units are astronomical units. Suppose a comet is following a path modeled by the equation \(x=y^{2}+20\) Will the comet necessarily hit Pluto? Explain.

Find the coordinates of the vertices and foci and the equations of the asymptotes for the hyperbola with the given equation. Then graph the hyperbola. $$ y^{2}-3 x^{2}+6 y+6 x-18=0 $$

Write an equation for each parabola described below. Then draw the graph. vertex \((0,1),\) focus \((0,5)\)

Find the center and radius of the circle with the given equation. Then graph the circle. $$ x^{2}+y^{2}+2 x-10=0 $$

Write an equation for the ellipse that satisfies each set of conditions. endpoints of minor axis at (0, 5) and (0, -5), foci at (12, 0) and (-12, 0)

ASTRONOMY For Exercises 27 and 28 , use the following information. The orbit of Pluto can be modeled by the equation \(\frac{x^{2}}{3951^{2}}+\frac{y^{2}}{38.3^{2}}=1,\) where the units are astronomical units. Suppose a comet is following a path modeled by the equation \(x=y^{2}+20 .\) Where do the graphs of \(y=2 x+1\) and \(2 x^{2}+y^{2}=11\) intersect?

Find the coordinates of the vertices and foci and the equations of the asymptotes for the hyperbola with the given equation. Then graph the hyperbola. $$ 4 x^{2}-25 y^{2}-8 x-96=0 $$

LIGHT For Exercises 29 and \(30,\) use the following information. A lamp standing near a wall throws an arc of light in the shape of a conic section. Suppose the edge of the light can be represented by the equation \(3 y^{2}-2 y-4 x^{2}+2 x-8=0\) . Identify the shape of the edge of the light.

Write an equation for each parabola described below. Then draw the graph. vertex \((8,6),\) focus \((2,6)\)

Find the center and radius of the circle with the given equation. Then graph the circle. $$ x^{2}+y^{2}-18 x-18 y+53=0 $$

Write an equation for the ellipse that satisfies each set of conditions. Write the equation \(10 x^{2}+2 y^{2}=40\) in standard form.

GEOMETRY A circle has a radius with endpoints at \((2,5)\) and \((-1,-4)\) . Find the circumference and area of the circle.

What are the coordinates of the points that lie on the graphs of both \(x^{2}+y^{2}=25\) and \(2 x^{2}+3 y^{2}=66 ?\)

Find an equation for a hyperbola centered at the origin with a horizontal transverse axis of length 8 units and a conjugate axis of length 6 units.

LIGHT For Exercises 29 and \(30,\) use the following information. A lamp standing near a wall throws an arc of light in the shape of a conic section. Suppose the edge of the light can be represented by the equation \(3 y^{2}-2 y-4 x^{2}+2 x-8=0\) . Graph the equation.

Write an equation for each parabola described below. Then draw the graph. focus \((-4,-2),\) directrix \(x=-8\)

Find the center and radius of the circle with the given equation. Then graph the circle. $$ x^{2}+y^{2}+9 x-8 y+4=0 $$

Write an equation for the ellipse that satisfies each set of conditions. What is the standard form of the equation \(x^{2}+6 y^{2}-2 x+12 y-23=0 ?\)

GEOMETRY Circle \(Q\) has a diameter \(\overline{A B}\) . If \(A\) is at \((-3,-5)\) and the center of the circle is at \((2,3),\) find the coordinates of \(B\) .

Rockers Two rockets are launched at the same time, but from different heights. The height \(y\) in feet of one recket after \(t\) seconds is given by \(y=-16 t^{2}+150 t+5 .\) The height of the other rocket is given by \(y=-16 t^{2}+\) 160\(t\) . After how many seconds are the rockets at the same height?

What is an equation for the hyperbola centered at the origin with a vertical transverse axis of length 12 units and a conjugate axis of length 4 units?

If two stones are thrown into a lake at different points, the points of intersection of the resulting ripples will follow a conic section. Suppose the conic section has the equation \(x^{2}-2 y^{2}-2 x-5=0\) Identify the shape of the curve.

Write an equation for each parabola described below. Then draw the graph. vertex \((1,7),\) directrix \(y=3\)

Find the center and radius of the circle with the given equation. Then graph the circle. $$ x^{2}+y^{2}-3 x+8 y=20 $$

GEOGRAPHY For Exercises 31 and \(32,\) use the following information. The U.S. Geological Survey (USGS) has determined the official center of the United States. Approximate the center of the United States. Describe vour method.

STRUCTURAL DESIGN An architect's design for a building includes some large pillars with cross sections in the shape of hyperbolas. The curves can be modeled by the equation \(\frac{x^{2}}{0.25}-\frac{y^{2}}{9}=1,\) where the units are in meters. If the pillars are 4 meters tall, find the width of the top of each pillar and the width of each pillar at the narrowest point in the middle. Round to the nearest centimeter.

If two stones are thrown into a lake at different points, the points of intersection of the resulting ripples will follow a conic section. Suppose the conic section has the equation \(x^{2}-2 y^{2}-2 x-5=0\) Graph the equation.

Write an equation for each parabola described below. Then draw the graph. vertex \((-7,4),\) axis of symmetry \(x=-7,\) measure of latus rectum \(6, a<0\)

Write an equation for the circle that satisfies each set of conditions. center \((8,-9),\) passes through \((21,22)\)

In an ellipse, the ratio \(\frac{c}{a}\) is called the eccentricity and is denoted by the letter \(e\) . Eccentricity measures the elongation of an ellipse. The closer \(e\) is to \(0,\) the more an ellipse looks like a circle. Pluto has the most eccentric orbit in our solar system with \(e \approx 0.25 .\) Find an equation to model the orbit of Pluto, given that the length of the major axis is about 7.34 billion miles. Assume that the major axis is horizontal and that the center of the orbit is the origin.

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. Solve each equation for \(y\)

A curved mirror is placed in a store for a wide-angle view of the room. The equation \(\frac{x^{2}}{1}-\frac{y^{2}}{3}=1\) models the curvature of the mirror. A small security camera is placed 3 feet from the vertex of the mirror so that a diameter of 2 feet of the mirror is visible. If the back of the room lies on \(x=-18\) , what width of the back of the room is visible to the camera?

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

Write an equation for the circle that satisfies each set of conditions. center \((-\sqrt{13}, 42),\) passes through the origin

Explain why a circle is a special case of an ellipse.

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. Plot some points and use them to graph the equation. Be sure to consider negative values for the variables.