Solved: 4346 describe procedures that are to be applied to numbers. In each exercise, a

In Exercises 1-8, match the equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), (f), (g), and (h).] \(y^2 = 4x\) Text Transcription: y^2 = 4x

In Exercises 1-8, match the equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), (f), (g), and (h).] \((x+4)^2=2(y+2)\) Text Transcription: (x+4)^2 = 2(y+2)

In Exercises 1-8, match the equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), (f), (g), and (h).] \((x+4)^2=-2(y-2)\) Text Transcription: (x+4)^2 = -2(y-2)

In Exercises 1-8, match the equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), (f), (g), and (h).] \(\frac{(x-2)^2}{16}+\frac{(y+1)^2}{4}=1\) Text Transcription: (x-2)^2 / 16 + (y+1)^2 / 4 = 1

In Exercises 1-8, match the equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), (f), (g), and (h).] \(\frac{x^2}{4}+\frac{y^2}{9}=1\) Text Transcription: x^2 / 4 + y^2 / 9 = 1

In Exercises 1-8, match the equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), (f), (g), and (h).] \(\frac{x^2}{16}+\frac{y^2}{16}=1\) Text Transcription: x^2 / 16 + y^2 /16 =1

In Exercises 1-8, match the equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), (f), (g), and (h).] \(\frac{y^2}{16}-\frac{x^2}{1}=1\) Text Transcription: y^2 /16 - x^2 /1 =1

In Exercises 1-8, match the equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), (f), (g), and (h).] \(\frac{(x-2)^2}{9}-\frac{y^2}{4}=1\) Text Transcription: (x-2)^2 / 9 - y^2 / 4 =1

In Exercises 9–16, find the vertex, focus, and directrix of the parabola, and sketch its graph. \(y^2=-8x\) Text Transcription: y^2 = -8x

In Exercises 9–16, find the vertex, focus, and directrix of the parabola, and sketch its graph. \(x^2+6y=0\) Text Transcription: x^2 + 6y = 0

In Exercises 9–16, find the vertex, focus, and directrix of the parabola, and sketch its graph. \((x+5)+(y-3)^2=0\) Text Transcription: (x+5) + (y-3)^2 = 0

In Exercises 9–16, find the vertex, focus, and directrix of the parabola, and sketch its graph. \((x-6)^2+8(y+7)=0\) Text Transcription: (x-6)^2 +8(y+7)=0

In Exercises 9–16, find the vertex, focus, and directrix of the parabola, and sketch its graph. \(y^2-4y-4x=0\) Text Transcription: y^2 -4y -4x = 0

In Exercises 9–16, find the vertex, focus, and directrix of the parabola, and sketch its graph. \(y^2+6y+8x+25=0\) Text Transcription: y^2 + 6y + 8x + 25 = 0

In Exercises 9–16, find the vertex, focus, and directrix of the parabola, and sketch its graph. \(x^2+4x+4y-4=0\) Text Transcription: x^2 +4x +4y -4=0

In Exercises 9–16, find the vertex, focus, and directrix of the parabola, and sketch its graph. \(y^2+4y+8x-12=0\) Text Transcription: y^2 +4y +8x -12 = 0

In Exercises 17–20, find the vertex, focus, and directrix of the parabola. Then use a graphing utility to graph the parabola. \(y^2+x+y=0\) Text Transcription: y^2 + x + y = 0

In Exercises 17–20, find the vertex, focus, and directrix of the parabola. Then use a graphing utility to graph the parabola. \(y=-\frac{1}{6}(x^2-8x+6)\) Text Transcription: y=-1/6 (x^2 -8x+6)

In Exercises 17–20, find the vertex, focus, and directrix of the parabola. Then use a graphing utility to graph the parabola. \(y^2-4x-4=0\) Text Transcription: y^2 -4x -4 =0

In Exercises 17–20, find the vertex, focus, and directrix of the parabola. Then use a graphing utility to graph the parabola. \(x^2-2x+8y+9=0\) Text Transcription: x^2 -2x + 8y + 9 = 0

In Exercises 21–28, find an equation of the parabola. Vertex: (5, 4) Focus: (3, 4)

In Exercises 21–28, find an equation of the parabola. Vertex: (-2, 1) Focus: (-2, -1)

In Exercises 21–28, find an equation of the parabola. Vertex: (0, 5) Directrix: y = -3

In Exercises 21–28, find an equation of the parabola. Focus: (2, 2) Directrix: x = -2

In Exercises 21–28, find an equation of the parabola.

In Exercises 21–28, find an equation of the parabola.

In Exercises 21–28, find an equation of the parabola. Axis is parallel to y-axis; graph passes through (0, 3), (3, 4), and (4, 11).

In Exercises 21–28, find an equation of the parabola. Directrix: y = -2; endpoints of latus rectum are (0, 2) and (8, 2).

In Exercises 29–34, find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph. \(16x^2+y^2=16\) Text Transcription: 16x^2 + y^2 = 16

In Exercises 29–34, find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph. \(3x^2+7y^2=63\) Text Transcription: 3x^2 + 7y^2 = 63

In Exercises 29–34, find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph. \(\frac{(x-3)^2}{16}+\frac{(y-1)^2}{25}=1\) Text Transcription: (x-3)^2 /16 + (y-1)^2 /25 =1

In Exercises 29–34, find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph. \((x+4)^2+\frac{(y+6)^2}{1/4}=1\) Text Transcription: (x+4)^2 +(y+6)^2 /1/4 =1

In Exercises 29–34, find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph. \(9x^2+4y^2+36x-24y+36=0\) Text Transcription: 9x^2 +4y^2 +36x -24y +36=0

In Exercises 29–34, find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph. \(16x^2+25y^2-64x+150y+279=0\) Text Transcription: 16x^2 +25y^2 -64x +150y +279 =0

In Exercises 35–38, find the center, foci, and vertices of the ellipse. Use a graphing utility to graph the ellipse. \(12x^2+20y^2-12x+40y-37=0\) Text Transcription: 12x^2 +20y^2 -12x +40y -37 =0

In Exercises 35–38, find the center, foci, and vertices of the ellipse. Use a graphing utility to graph the ellipse. \(36x^2+9y^2+48x-36y+43=0\) Text Transcription: 36x^2 +9y^2 +48x -36y +43 =0

In Exercises 35–38, find the center, foci, and vertices of the ellipse. Use a graphing utility to graph the ellipse. \(x^2+2y^2-3x+4y+0.25=0\) Text Transcription: x^2 +2y^2 -3x+4y+0.25=0

In Exercises 35–38, find the center, foci, and vertices of the ellipse. Use a graphing utility to graph the ellipse. \(2x^2+y^2+4.8x-6.4y+3.12=0\) Text Transcription: 2x^2 +y^2 +4.8x -6.4y +3.12 =0

In Exercises 39–44, find an equation of the ellipse. Center: (0, 0) Focus: (5, 0) Vertex: (6, 0)

In Exercises 39–44, find an equation of the ellipse. Vertices: (0, 3), (8, 3) Eccentricity: \(\frac{3}{4}\) Text Transcription: 3/4

In Exercises 39–44, find an equation of the ellipse. Vertices: (3, 1), (3, 9) Minor axis length: 6

In Exercises 39–44, find an equation of the ellipse. Foci: \((0,\ \pm9)\) Major axis length: 22 Text Transcription: (0, pm 9)

In Exercises 39–44, find an equation of the ellipse. Center: (0, 0) Major axis: horizontal Points on the ellipse: (3, 1), (4, 0)

In Exercises 39–44, find an equation of the ellipse. Center: (1, 2) Major axis: vertical Points on the ellipse: (1, 6), (3, 2)

In Exercises 45–52, find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid. \(y^2-\frac{x^2}{9}=1\) Text Transcription: y^2 -x^2 /9 =1

In Exercises 45–52, find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid. \(\frac{x^2}{25}-\frac{y^2}{16}=1\) Text Transcription: x^2 /25 - y^2 /16 =1

In Exercises 45–52, find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid. \(\frac{(x-1)^2}{4}-\frac{(y+2)^2}{1}=1\) Text Transcription: (x-1)^2 /4 - (y+2)^2 /1 =1

In Exercises 45–52, find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid. \(\frac{(y+3)^2}{225}-\frac{(x-5)^2}{64}=1\) Text Transcription: (y+3)^2 /225 - (x-5)^2 /64 =1

In Exercises 45–52, find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid. \(9x^2-y^2-36x-6y+18=0\) Text Transcription: 9x^2 -y^2 -36x -6y +18 =0

In Exercises 45–52, find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid. \(y^2-16x^2+64x-208=0\) Text Transcription: y^2 -16x^2 +64x -208 =0

In Exercises 45–52, find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid. \(x^2-9y^2+2x-54y-80=0\) Text Transcription: x^2 -9y^2 +2x -54y -80 =0

In Exercises 45–52, find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid. \(9x^2-4y^2+54x+8y+78=0\) Text Transcription: 9x^2 -4y^2 +54x+8y+78=0

In Exercises 53 –56, find the center, foci, and vertices of the hyperbola. Use a graphing utility to graph the hyperbola and its asymptotes. \(9y^2-x^2+2x+54y+62=0\) Text Transcription: 9y^2 -x^2 +2x +54y+62 =0

In Exercises 53 –56, find the center, foci, and vertices of the hyperbola. Use a graphing utility to graph the hyperbola and its asymptotes. \(9x^2-y^2+54x+10y+55=0\) Text Transcription: 9x^2 -y^2 +54x +10y +55 =0

In Exercises 53 –56, find the center, foci, and vertices of the hyperbola. Use a graphing utility to graph the hyperbola and its asymptotes. \(3x^2-2y^2-6x-12y-27=0\) Text Transcription: 3x^2 -2y^2 -6x-12y-27=0

In Exercises 53 –56, find the center, foci, and vertices of the hyperbola. Use a graphing utility to graph the hyperbola and its asymptotes. \(3y^2-x^2+6x-12y=0\) Text Transcription: 3y^2 -x^2 +6x-12y=0

In Exercises 57– 64, find an equation of the hyperbola. Vertices: \((\pm1,\ 0)\) Asymptotes: \(y = \pm5x\) Text Transcription: (pm1, 0) y=pm5x)

In Exercises 57– 64, find an equation of the hyperbola. Vertices: (0,\ \pm4) Asymptotes: \(y = \pm2x\) Text Transcription: (0, pm4) (y=pm2x)

In Exercises 57– 64, find an equation of the hyperbola. Vertices: \((2,\ \pm3)\) Point on graph: (0, 5) Text Transcription: (2, pm3)

In Exercises 57– 64, find an equation of the hyperbola. Vertices: \((2,\ \pm3)\) Foci: \((2,\ \pm5)\) Text Transcription: (2, pm3) (2, pm5)

In Exercises 57– 64, find an equation of the hyperbola. Center: (0, 0) Vertex: (0, 2) Focus: (0, 4)

In Exercises 57– 64, find an equation of the hyperbola. Center: (0, 0) Vertex: (6, 0) Focus: (10, 0)

In Exercises 57– 64, find an equation of the hyperbola. Vertices: (0, 2), (6, 2) Asymptotes: \(y=\frac{2}{3}x\) \(y=4-\frac{2}{3}x\) Text Transcription: y=2/3 x y=4-2/3 x

In Exercises 57– 64, find an equation of the hyperbola. Focus: (20, 0) Asymptotes: \(y=\pm\frac{3}{4}x\) Text Transcription: y=pm 3/4 x

In Exercises 65 and 66, find equations for (a) the tangent lines and (b) the normal lines to the hyperbola for the given value of x. \(\frac{x^2}{9}-y^2=1,\ \ \ x=6\) Text Transcription: x^2 /9 -y^2 =1, x=6

In Exercises 65 and 66, find equations for (a) the tangent lines and (b) the normal lines to the hyperbola for the given value of x. \(\frac{y^2}{4}-\frac{x^2}{2}=1,\ \ \ x=4\) Text Transcription: y^2 /4 - x^2 /2 =1, x = 4

In Exercises 67–76, classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. \(x^2 +4y^2-6x+16y+21=0\) Text Transcription: x^2 +4y^2 -6x +16y +21 =0

In Exercises 67–76, classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. \(4x^2-y^2-4x-3=0\) Text Transcription: 4x^2 -y^2 -4x-3=0

In Exercises 67–76, classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. \(y^2-8y-8x=0\) Text Transcription: y^2 -8y-8x=0

In Exercises 67–76, classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. \(25x^2-10x-200y-119=0\) Text Transcription: 25x^2 -10x -200y -119 =0

In Exercises 67–76, classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. \(4x^2+4y^2-16y+15=0\) Text Transcription: 4x^2 +4y^2 -16y+15=0

In Exercises 67–76, classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. \(y^2-4y=x+5\) Text Transcription: y^2 -4y = x+5

In Exercises 67–76, classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. \(9x^2+9y^2-36x+6y+34=0\) Text Transcription: 9x^2 +9y^2 -36x+6y+34=0

In Exercises 67–76, classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. 2x(x - y) = y(3 - y - 2x)

In Exercises 67–76, classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. \(3(x-1)^2=6+2(y+1)^2\) Text Transcription: 3(x - 1)^2 = 6 + 2(y + 1)^2

In Exercises 67–76, classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. \(9(x+3)^2=36-4(y-2)^2\) Text Transcription: 9(x+3)^2 = 36 - 4(y - 2)^2

(a) Give the definition of a parabola. (b) Give the standard forms of a parabola with vertex at (h, k). (c) In your own words, state the reflective property of a parabola.

(a) Give the definition of an ellipse. (b) Give the standard forms of an ellipse with center at (h, k).

(a) Give the definition of a hyperbola. (b) Give the standard forms of a hyperbola with center at (h, k). (c) Write equations for the asymptotes of a hyperbola.

Define the eccentricity of an ellipse. In your own words, describe how changes in the eccentricity affect the ellipse.

Solar Collector A solar collector for heating water is constructed with a sheet of stainless steel that is formed into the shape of a parabola (see figure). The water will flow through a pipe that is located at the focus of the parabola. At what distance from the vertex is the pipe?

Beam Deflection A simply supported beam that is 16 meters long has a load concentrated at the center (see figure). The deflection of the beam at its center is 3 centimeters. Assume that the shape of the deflected beam is parabolic. (a) Find an equation of the parabola. (Assume that the origin is at the center of the beam.) (b) How far from the center of the beam is the deflection 1 centimeter?

Find an equation of the tangent line to the parabola \(y=ax^2\ \text{at}\ x=x_0\). Prove that the x-intercept of this tangent line is \((x_0/2,\ 0)\). Text Transcription: y=ax^2 at x=x_0 (x_0 /2, 0)

(a) Prove that any two distinct tangent lines to a parabola intersect. (b) Demonstrate the result of part (a) by finding the point of intersection of the tangent lines to the parabola \(x^2-4x-4y=0\) at the points (0, 0) and (6, 3). Text Transcription: x^2 -4x -4y =0

(a) Prove that if any two tangent lines to a parabola intersect at right angles, their point of intersection must lie on the directrix. (b) Demonstrate the result of part (a) by proving that the tangent lines to the parabola \(x^2-4x-4y+8=0\) at the points (-2, 5) and \((3,\ \frac{5}{4})\) intersect all right angles, and that the point of intersection lies on the directrix. Text Transcription: x^2 -4x-4y+8=0 (3, 5/4)

Find the point of the graph of \(x^2=8y\) that is closest to the focus of the parabola.

Radio and Television Reception In mountainous areas, reception of radio and television is sometimes poor. Consider an idealized case where a hill is represented by the graph of the parabola \(y=x-x^2\), a transmitter is located at the point (-1, 1), and a receiver is located on the other side of the hill at the point \((x_0,\ 0)\). What is the closest the receiver can be to the hill while still maintaining unobstructed reception? Text Transcription: y=x-x^2 (x_0 , 0)

Modeling Data The table shows the average amount of time A (in minutes) women spent watching television each day for the years 1999 through 2005. (Source: Nielsen Media Research) (a) Use the regression capabilities of a graphing utility to find a model of the form \(A=at^2+bt+c\) for the data. Let t represent the year, with t = 9 corresponding to 1999. (b) Use a graphing utility to plot the data and graph the model. (c) Find dA/dt and sketch its graph for \(9\ \leq\ t\ \leq\ 15\). What information about the average amount of time women spend watching television is given by the graph of the derivative? Text Transcription: A=at^2 + bt + c 9 leq t leq 15

Architecture A church window is bounded above by a parabola and below by the arc of a circle (see figure). Find the surface area of the window.

Arc Length Find the arc length of the parabola \(4x-y^2=0\) over the interval \(0\ \leq\ y\ \leq\ 4\). Text Transcription: 4x-y^2 =0 0 leq y leq 4

Bridge Design A cable of a suspension bridge is suspended (in the shape of a parabola) between two towers that are 120 meters apart and 20 meters above the roadway (see figure). The cables touch the roadway midway between the towers. (a) Find an equation for the parabolic shape of each cable. (b) Find the length of the parabolic supporting cable.

Surface Area A satellite signal receiving dish is formed by revolving the parabola given by \(x^2=20y\) about the y-axis. The radius of the dish is r feet. Verify that the surface area of the dish is given by \(2\pi \int _0 ^r\ x\sqrt{1+(\frac{x}{10})^2}\ dx=\frac{\pi}{15}[(100+r^2)^{3/2}-1000]\). Text Transcription: x^2 =20y 2pi int _0 ^r x sqrt1+(x/10)^2 dx = pi/15 [(100+r^2)^3/2 -1000]

Investigation Sketch the graphs of \(x^2=4py\) for \(p=\frac{1}{4},\ \frac{1}{2},\ 1,\ \frac{3}{2}\) and 2 on the same coordinate axes. Discuss the change in the graphs as p increases. Text Transcription: x^2 =4py p=1/4 , 1/2 , 1, 3/2

Area Find a formula for the area of the shaded region in the figure.

Writing On page 699, it was noted that an ellipse can be drawn using two thumbtacks, a string of fixed length (greater than the distance between the tacks), and a pencil. If the ends of the string are fastened at the tacks and the string is drawn taut with a pencil, the path traced by the pencil will be an ellipse. (a) What is the length of the string in terms of a? (b) Explain why the path is an ellipse.

Construction of a Semielliptical Arch A fireplace arch is to be constructed in the shape of a semi ellipse. The opening is to have a height of 2 feet at the center and a width of 5 feet along the base (see figure). The contractor draws the outline of the ellipse by the method shown in Exercise 95. Where should the tacks be placed and what should be the length of the piece of string?

Sketch the ellipse that consists of all points (x, y) such that the sum of the distances between (x, y) and two fixed points is 16 units, and the foci are located at the centers of the two sets of concentric circles in the figure. To print an enlarged copy of the graph, go to the website www.mathgraphs.com.

Orbit of Earth Earth moves in an elliptical orbit with the sun at one of the foci. The length of half of the major axis is 149,598,000 kilometers, and the eccentricity is 0.0167. Find the minimum distance (perihelion) and the maximum distance (aphelion) of Earth from the sun.

Satellite Orbit The apogee (the point in orbit farthest from Earth) and the perigee (the point in orbit closest to Earth) of an elliptical orbit of an Earth satellite are given by and Show that the eccentricity of the orbit is \(e=\frac{A-P}{A+P}\) Text Transcription: e=A-P/A+P

Explorer 18 On November 27, 1963, the United States launched the research satellite Explorer 18. Its low and high points above the surface of Earth were 119 miles and 123,000 miles. Find the eccentricity of its elliptical orbit.

Explorer 55 On November 20, 1975, the United States launched the research satellite Explorer 55. Its low and high points above the surface of Earth were 96 miles and 1865 miles. Find the eccentricity of its elliptical orbit.

Consider the equation \(9x^2+4y^2-36x-24y-36=0\). (a) Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. (b) Change the \(4y^2\)-term in the equation to \(-4y^2\). Classify the graph of the new equation. (c) Change the \(9x^2\)-term in the original equation to \(4x^2\)/ Classify the graph of the new equation. (d) Describe one way you could change the original equation so that its graph is a parabola. Text Transcription: 9x^2 +4y^2 -36x-24y-36=0 4y^2 -4y^2 9x^2 4x^2

Halley’s Comet Probably the most famous of all comets, Halley’s comet, has an elliptical orbit with the sun at one focus. Its maximum distance from the sun is approximately 35.29 AU (1 astronomical unit \(\approx\ 92.956\ \times\ 10^6\) miles), and its minimum distance is approximately 0.59 AU. Find the eccentricity of the orbit. Text Transcription: approx 92.956 times 10^6

The equation of an ellipse with its center at the origin can be written as \(\frac{x^2}{a^2}+\frac{y^2}{a^2(1-e^2)}=1\). Show that as \(e\rightarrow 0\), with a remaining fixed, the ellipse approaches a circle. Text Transcription: x^2 /a^2 + y^2 /a^2 (1-e^2) =1 e rightarrow 0

Consider a particle traveling clockwise on the elliptical path \(\frac{x^2}{100}+\frac{y^2}{25}=1\). The particle leaves the orbit at the point (-8, 3) and travels in a straight line tangent to the ellipse. At what point will the particle cross the y-axis? Text Transcription: x^2 /100 + y^2 /25 =1

Volume The water tank on a fire truck is 16 feet long, and its cross sections are ellipses. Find the volume of water in the partially filled tank as shown in the figure.

In Exercises 107 and 108, determine the points at which dy/dx is zero or does not exist to locate the endpoints of the major and minor axes of the ellipse. \(16x^2+9y^2+96x+36y+36=0\) Text Transcription: 16x^2 +9y^2 +96x +36y +36=0

In Exercises 107 and 108, determine the points at which dy/dx is zero or does not exist to locate the endpoints of the major and minor axes of the ellipse. \(9x^2+4y^2+36x-24y+36=0\) Text Transcription: 9x^2 +4y^2 +36x -24y+36=0

Area and Volume In Exercises 109 and 110, find (a) the area of the region bounded by the ellipse, (b) the volume and surface area of the solid generated by revolving the region about its major axis (prolate spheroid), and (c) the volume and surface area of the solid generated by revolving the region about its minor axis (oblate spheroid). \(\frac{x^2}{4}+\frac{y^2}{1}=1\) Text Transcription: x^2 /4 +y^2 /1 =1

Area and Volume In Exercises 109 and 110, find (a) the area of the region bounded by the ellipse, (b) the volume and surface area of the solid generated by revolving the region about its major axis (prolate spheroid), and (c) the volume and surface area of the solid generated by revolving the region about its minor axis (oblate spheroid). \(\frac{x^2}{16}+\frac{y^2}{9}=1\) Text Transcription: x^2 /16 +y^2 /9 =1

Arc Length Use the integration capabilities of a graphing utility to approximate to two-decimal-place accuracy the elliptical integral representing the circumference of the ellipse \(\frac{x^2}{25}+\frac{y^2}{49}=1\). Text Transcription: x^2 /25 +y^2 /49 =1

Prove Theorem 10.4 by showing that the tangent line to an ellipse at a point P makes equal angles with lines through P and the foci (see figure). [Hint: (1) Find the slope of the tangent line at P, (2) find the slopes of the lines through P and each focus, and (3) use the formula for the tangent of the angle between two lines.]

Geometry The area of the ellipse in the figure is twice the area of the circle. What is the length of the major axis?

Conjecture (a) Show that the equation of an ellipse can be written as \(\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{a^2(1-e^2)}=1\). (b) Use a graphing utility to graph the ellipse \(\frac{(x-2)^2}{4}+\frac{(y-3)^2}{4(1-e^2)}=1\). (c) Use the results of part (b) to make a conjecture about the change in the shape of the ellipse as e approaches 0. Text Transcription: (x-h)^2 /a^2 +(y-k)^2 /a^2(1-e^2)=1 (x-2)^2 /4 + (y-3)^2 /4(1-e^2) =1

Find an equation of the hyperbola such that for any point on the hyperbola, the difference between its distances from the points (2, 2) and (10, 2) is 6.

Find an equation of the hyperbola such that for any point on the hyperbola, the difference between its distances from the points (-3, 0) and (-3, 3) is 2.

Sketch the hyperbola that consists of all points (x, y) such that the difference of the distances between (x, y) and two fixed points is 10 units, and the foci are located at the centers of the two sets of concentric circles in the figure. To print an enlarged copy of the graph, go to the website www.mathgraphs.com.

Consider a hyperbola centered at the origin with a horizontal transverse axis. Use the definition of a hyperbola to derive its standard form: \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\). Text Transcription: x^2 /a^2 -y^2 /b^2 =1

Sound Location A rifle positioned at point (-c, 0) is fired at a target positioned at point (c, 0). A person hears the sound of the rifle and the sound of the bullet hitting the target at the same time. Prove that the person is positioned on one branch of the hyperbola given by \(\frac{x^2}{c^2v_s^2/v^2m}-\frac{y^2}{c^2(v_m^2-v_s^2)/v_m^2}=1\) where \(v_m\) is the muzzle velocity of the rifle and \(v_s\) is the speed of sound, which is about 1100 feet per second. Text Transcription: x^2 /c^2 v_s ^2 / v_m ^2 - y^2 / c^2(v_m ^2 - v_s ^2) v_m ^2 =1 v_m v_s

Navigation LORAN (long distance radio navigation) for aircraft and ships uses synchronized pulses transmitted by widely separated transmitting stations. These pulses travel at the speed of light (186,000 miles per second). The difference in the times of arrival of these pulses at an aircraft or ship is constant on a hyperbola having the transmitting stations as foci. Assume that two stations, 300 miles apart, are positioned on a rectangular coordinate system at (-150, 0) and (150, 0) and that a ship is traveling on a path with coordinates (x, 75) (see figure). Find the x-coordinate of the position of the ship if the time difference between the pulses from the transmitting stations is 1000 microseconds (0.001 second).

Hyperbolic Mirror A hyperbolic mirror (used in some telescopes) has the property that a light ray directed at the focus will be reflected to the other focus. The mirror in the figure has the equation \((x^2/36)-(y^2/64)=1\). At which point on the mirror will light from the point (0, 10) be reflected to the other focus? Text Transcription: (x^2 /36) - (y^2 /64)=1

Show that the equation of the tangent line to \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) at the point \((x_0,\ y_0)\) is \((x_0 /a^2)x-(y_0 /b^2)y=1\). Text Transcription: x^2 /a^2 -y^2 /b^2 =1 (x_0 , y_0) (x_0/a^2)x-(y_0/b^2)y=1

Show that the graphs of the equations intersect at right angles: \(\frac{x^2}{a^2}+\frac{2y^2}{b^2}=1\) and \(\frac{x^2}{a^2 -b^2}-\frac{2y^2}{b^2}=1\). Text Transcription: x^2 /a^2 + 2y^2 /b^2 =1 x^2 /a^2 -b^2 -2y^2 /b^2 =1

Prove that the graph of the equation \(Ax^2+Cy^2+Dx+Ey+F=0\) is one of the following (except in degenerate cases). Conic Condition (a) Circle A = C (b) Parabola A = 0 or C = 0 (but not both) (c) Ellipse AC > 0 (d) Hyperbola AC < 0 Text Transcription: Ax^2 +Cy^2 +Dx +Ey + F = 0

True or False? In Exercises 125–130, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. It is possible for a parabola to intersect its directrix.

True or False? In Exercises 125–130, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The point on a parabola closest to its focus is its vertex.

True or False? In Exercises 125–130, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If C is the circumference of the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,\ \ \ b\ <\ a\) then \(2\pi b\ \leq\ C\ \leq\ 2\pi a\). Text Transcription: x^2 /a^2 +y^2 /b^2 =1, b < a 2pi b leq C leq 2pi a

True or False? In Exercises 125–130, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(D\ \neq\ 0\ \text{or}\ E\ \neq\ 0\), then the graph of \(y^2-x^2+Dx+Ey=0\) is a hyperbola. Text Transcription: D neq 0 or E neq 0 y^2 -x^2 +Dx +Ey =0

True or False? In Exercises 125–130, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If the asymptotes of the hyperbola \((x^2 /a^2)-(y^2 /b^2)=1\) intersect at right angles, then a = b. Text Transcription: (x^2 /a^2) - (y^2 /b^2) = 1

True or False? In Exercises 125–130, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Every tangent line to a hyperbola intersects the hyperbola only at the point of tangency.

For a point P on an ellipse, let d be the distance from the center of the ellipse to the line tangent to the ellipse at P. Prove that \((PF_1)(PF_2)d^2\) is constant as P varies on the ellipse, where \(PF_1\ \text{and}\ PF_2\) are the distances from P to the foci \(F_1\ \text{and}\ F_2\) of the ellipse. Text Transcription: (PF_1)(PF_2)d^2 PF_1 and PF_2 F_1 and F_2

Find the minimum value of \((u-2)^2+(\sqrt{2-u^2}-\frac{9}{v})^2\) for \(0\ <\ u\ <\ \sqrt{2}\ \text{and}\ v\ >\ 0\). Text Transcription: (u-v)^2 + (sqrt2-u^2 -9/v)^2 0 < u < sqrt2 and v > 0