Calculus of a Single Variable

Ron Larson, Bruce Edwards

Chapter 10

Conics, Parametric Equations, and Polar Coordinates - all with Video Answers

Educators

Section 1

Conics and Calculus

Problem 1

Matching In Exercises $1-6,$ match the equation with its graph. [The graphs are labeled (a), (b), (c), (e), and (f).
$$
y^{2}=4 x
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 2

Matching In Exercises $1-6,$ match the equation with its graph. [The graphs are labeled (a), (b), (c), (e), and (f).
$$
(x+4)^{2}=-2(y-2)
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 3

Matching In Exercises $1-6,$ match the equation with its graph. [The graphs are labeled (a), (b), (c), (e), and (f).
$$
\frac{y^{2}}{16}-\frac{x^{2}}{1}=1
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 4

Matching In Exercises $1-6,$ match the equation with its graph. [The graphs are labeled (a), (b), (c), (e), and (f).
$$
\frac{(x-2)^{2}}{16}+\frac{(y+1)^{2}}{4}=1
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 5

Matching In Exercises $1-6,$ match the equation with its graph. [The graphs are labeled (a), (b), (c), (e), and (f).
$$
\frac{x^{2}}{4}+\frac{y^{2}}{9}=1
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 6

Matching In Exercises $1-6,$ match the equation with its graph. [The graphs are labeled (a), (b), (c), (e), and (f).
$$
\frac{(x-2)^{2}}{9}-\frac{y^{2}}{4}=1
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 7

Sketching a Parabola In Exercises $7-14$ , find the vertex, focus, and directrix of the parabola, and sketch its graph.
$$
y^{2}=-8 x
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 8

Sketching a Parabola In Exercises $7-14$ , find the vertex, focus, and directrix of the parabola, and sketch its graph.
$$
x^{2}+6 y=0
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 9

Sketching a Parabola In Exercises $7-14$ , find the vertex, focus, and directrix of the parabola, and sketch its graph.
$$
(x+5)+(y-3)^{2}=0
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 10

Sketching a Parabola In Exercises $7-14$ , find the vertex, focus, and directrix of the parabola, and sketch its graph.
$$
(x-6)^{2}+8(y+7)=0
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 11

Sketching a Parabola In Exercises $7-14$ , find the vertex, focus, and directrix of the parabola, and sketch its graph.
$$
y^{2}-4 y-4 x=0
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 12

Sketching a Parabola In Exercises $7-14$ , find the vertex, focus, and directrix of the parabola, and sketch its graph.
$$
y^{2}+6 y+8 x+25=0
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 13

Sketching a Parabola In Exercises $7-14$ , find the vertex, focus, and directrix of the parabola, and sketch its graph.
$$
x^{2}+4 x+4 y-4=0
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 14

Sketching a Parabola In Exercises $7-14$ , find the vertex, focus, and directrix of the parabola, and sketch its graph.
$$
y^{2}+4 y+8 x-12=0
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 15

Finding an Equation of a Parabola In Exercises $15-22$ , find an equation of the parabola.
Vertex: $(5,4)$
Focus: $(3,4)$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 16

Finding an Equation of a Parabola In Exercises $15-22$ , find an equation of the parabola.
Vertex: $(-2,1)$
Focus: $(-2,-1)$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 17

Finding an Equation of a Parabola In Exercises $15-22$ , find an equation of the parabola.
Vertex: $(0,5)$
Directrix: $y=-3$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 18

Finding an Equation of a Parabola In Exercises $15-22$ , find an equation of the parabola.
Focus: $(2,2)$
Directrix: $x=-2$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 19

Finding an Equation of a Parabola In Exercises $15-22$ , find an equation of the parabola.
Vertex: $(0,4)$
Points on the parabola:
$(-2,0),(2,0)$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 20

Vertex: $(2,4)$
Points on the parabola:
$(0,0),(4,0)$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 21

Finding an Equation of a Parabola In Exercises $15-22$ , find an equation of the parabola.
Axis is parallel to $y$ -axis; graph passes through $(0,3),(3,4),$ and $(4,11) .$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 22

Finding an Equation of a Parabola In Exercises $15-22$ , find an equation of the parabola.
Directrix: $y=-2 ;$ endpoints of latus rectum are $(0,2)$ and $(8,2) .$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 23

Sketching an Ellipse In Exercises $23-28$ , find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph.
$$
16 x^{2}+y^{2}=16
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 24

Sketching an Ellipse In Exercises $23-28$ , find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph.
$$
3 x^{2}+7 y^{2}=63
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 25

Sketching an Ellipse In Exercises $23-28$ , find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph.
$$
\frac{(x-3)^{2}}{16}+\frac{(y-1)^{2}}{25}=1
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 26

Sketching an Ellipse In Exercises $23-28$ , find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph.
$$
(x+4)^{2}+\frac{(y+6)^{2}}{1 / 4}=1
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 27

Sketching an Ellipse In Exercises $23-28$ , find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph.
$$
9 x^{2}+4 y^{2}+36 x-24 y+36=0
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 28

Sketching an Ellipse In Exercises $23-28$ , find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph.
$$
16 x^{2}+25 y^{2}-64 x+150 y+279=0
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 29

Finding an Equation of an Ellipse In Exercises $29-34,$ find an equation of the ellipse.
Center: $(0,0)$
Focus: $(5,0)$
Vertex: $(6,0)$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 30

Finding an Equation of an Ellipse In Exercises $29-34,$ find an equation of the ellipse.
Vertices: $(0,3),(8,3)$
Eccentricity: $\frac{3}{4}$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 31

Finding an Equation of an Ellipse In Exercises $29-34,$ find an equation of the ellipse.
Vertices: $(3,1),(3,9)$
Minor axis length: 6

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 32

Finding an Equation of an Ellipse In Exercises $29-34,$ find an equation of the ellipse.
Foci: $(0, \pm 9)$
Major axis length: 22

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 33

Finding an Equation of an Ellipse In Exercises $29-34,$ find an equation of the ellipse.
Center: $(0,0)$
Major axis: horizontal
Points on the ellipse:
$(3,1),(4,0)$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 34

Finding an Equation of an Ellipse In Exercises $29-34,$ find an equation of the ellipse.
Center: $(1,2)$
Major axis: vertical
Points on the ellipse:
$(1,6),(3,2)$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 35

Sketching a Hyperbola In Exercises $35-40$ , find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid.
$$
\frac{x^{2}}{25}-\frac{y^{2}}{16}=1
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 36

Sketching a Hyperbola In Exercises $35-40$ , find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid.
$$
\frac{(y+3)^{2}}{225}-\frac{(x-5)^{2}}{64}=1
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 37

Sketching a Hyperbola In Exercises $35-40$ , find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid.
$$
9 x^{2}-y^{2}-36 x-6 y+18=0
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 38

Sketching a Hyperbola In Exercises $35-40$ , find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid.
$$
y^{2}-16 x^{2}+64 x-208=0
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 39

Sketching a Hyperbola In Exercises $35-40$ , find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid.
$$
x^{2}-9 y^{2}+2 x-54 y-80=0
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 40

Sketching a Hyperbola In Exercises $35-40$ , find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid.
$$
9 x^{2}-4 y^{2}+54 x+8 y+78=0
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 41

Finding an Equation of a Hyperbola In Exercises $41-48,$ find an equation of the hyperbola.
Vertices: $( \pm 1,0)$
Asymptotes: $y=\pm 5 x$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 42

Finding an Equation of a Hyperbola In Exercises $41-48,$ find an equation of the hyperbola.
Vertices: $(0, \pm 4)$
Asymptotes: $y=\pm 2 x$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 43

Finding an Equation of a Hyperbola In Exercises $41-48,$ find an equation of the hyperbola.
Vertices: $(2, \pm 3)$
Point on graph: $(0,5)$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 44

Finding an Equation of a Hyperbola In Exercises $41-48,$ find an equation of the hyperbola.
Vertices: $(2, \pm 3)$
Foci: $(2, \pm 5)$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 45

Finding an Equation of a Hyperbola In Exercises $41-48,$ find an equation of the hyperbola.
Center: $(0,0)$
Vertex: $(0,2)$
Focus: $(0,4)$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 46

Finding an Equation of a Hyperbola In Exercises $41-48,$ find an equation of the hyperbola.
Center: $(0,0)$
Vertex: $(6,0)$
Focus: $(10,0)$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 47

Finding an Equation of a Hyperbola In Exercises $41-48,$ find an equation of the hyperbola.
Vertices: $(0,2),(6,2)$
Asymptotes: $y=\frac{2}{3} x$

$$
y=4-\frac{2}{3} x
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 48

Finding an Equation of a Hyperbola In Exercises $41-48,$ find an equation of the hyperbola.
Focus: $(20,0)$
Asymptotes: $y=\pm \frac{3}{4} x$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 49

Finding Equations of Tangent Lines and Normal Lines In Exercises 49 and 50 , find equations for (a) the tangent lines and (b) the normal lines to the hyperbola for the given value of $x .$
$$
\frac{x^{2}}{9}-y^{2}=1, \quad x=6
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 50

Finding Equations of Tangent Lines and Normal Lines In Exercises 49 and 50 , find equations for (a) the tangent lines and (b) the normal lines to the hyperbola for the given value of $x .$
$$
\frac{y^{2}}{4}-\frac{x^{2}}{2}=1, \quad x=4
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 51

Classifying the Graph of an Equation In Exercises $51-58$ , classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.
$$
x^{2}+4 y^{2}-6 x+16 y+21=0
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 52

Classifying the Graph of an Equation In Exercises $51-58$ , classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.
$$
4 x^{2}-y^{2}-4 x-3=0
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 53

Classifying the Graph of an Equation In Exercises $51-58$ , classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.
$$
25 x^{2}-10 x-200 y-119=0
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 54

Classifying the Graph of an Equation In Exercises $51-58$ , classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.
$$
y^{2}-4 y=x+5
$$

Matthew Lee

Numerade Educator

Problem 55

Classifying the Graph of an Equation In Exercises $51-58$ , classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.
$$
9 x^{2}+9 y^{2}-36 x+6 y+34=0
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 56

Classifying the Graph of an Equation In Exercises $51-58$ , classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.
$$
2 x(x-y)=y(3-y-2 x)
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 57

Classifying the Graph of an Equation In Exercises $51-58$ , classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.
$$
3(x-1)^{2}=6+2(y+1)^{2}
$$

Brian Austin

Numerade Educator

Problem 58

Classifying the Graph of an Equation In Exercises $51-58$ , classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.
$$
9(x+3)^{2}=36-4(y-2)^{2}
$$

Matthew Lee

Numerade Educator

Problem 59

Parabola
(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.

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 60

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

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 61

Hyperbola
(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.

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 62

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

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 63

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

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 64

HOW DO YOU SEE IT? In parts (a)- -(d), describe in words how a plane could intersect with the double-napped cone to form the conic section (see figure).
$$
\begin{array}{ll}{\text { (a) Circle }} & {\text { (b) Ellipse }} \\ {\text { (c) Parabola }} & {\text { (d) Hyperbola }}\end{array}
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 65

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?

Regina Hays

Numerade Educator

Problem 66

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?

Matthew Lee

Numerade Educator

Problem 67

Proof
(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 $x^{2}-4 x-4 y=0$ at the points $(0,0)$ and $(6,3) .$

Regina Hays

Numerade Educator

Problem 68

Proof
(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 showing that the tangent lines to the parabola $x^{2}-4 x-4 y+8=0$ at the points $(-2,5)$ and $\left(3, \frac{5}{4}\right)$ intersect at right angles, and that the point of intersection lies on the directrix.

Regina Hays

Numerade Educator

Problem 69

Investigation Sketch the graphs of $x^{2}=4 p y$ 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.

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 70

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 cable touches the roadway midway between the towers.

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 71

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

Rowan Ahmed

Numerade Educator

Problem 72

Surface Area A satellite signal receiving dish is formed by revolving the parabola given by $x^{2}=20 y$ about the $y$ -axis. The radius of the dish is $r$ feet. Verify that the surface area of the dish is given by

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 73

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.

Regina Hays

Numerade Educator

Problem 74

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 $A$ and $P$ . Show that the eccentricity of the orbit is $$
e=\frac{A-P}{A+P}
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 75

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.

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 76

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.

Matthew Lee

Numerade Educator

Problem 77

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 $\mathrm{AU}$ (1 astronomical unit
is approximately $92.956 \times 10^{6}$ miles), and its minimum distance is approximately O.S9 AU. Find the eccentricity of the orbit.

Regina Hays

Numerade Educator

Problem 78

Particle Motion 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?

Matthew Lee

Numerade Educator

Problem 79

Area, Volume, and Surface Area In Exercises 79 and 80 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).
$$
\frac{x^{2}}{4}+\frac{y^{2}}{1}=1
$$

Monica Miller

Numerade Educator

Problem 80

Area, Volume, and Surface Area In Exercises 79 and 80 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).
$$
\frac{x^{2}}{16}+\frac{y^{2}}{9}=1
$$

Regina Hays

Numerade Educator

Problem 81

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
$$
\frac{x^{2}}{25}+\frac{y^{2}}{49}=1
$$

Regina Hays

Numerade Educator

Problem 82

Conjecture
(a) Show that the equation of an ellipse can be written as
$$
\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{a^{2}\left(1-e^{2}\right)}=1
$$
(b) Use a graphing utility to graph the ellipse
$$\frac{(x-2)^{2}}{4}+\frac{(y-3)^{2}}{4\left(1-e^{2}\right)}=1$$
for $e=0.95, e=0.75, e=0.5, e=0.25,$ and $e=0$
(c) Use the results of part (b) to make a conjecture about the change in the shape of the ellipse as $e$ approaches $0 .$

Matthew Lee

Numerade Educator

Problem 83

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

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 84

Proof 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.]

Regina Hays

Numerade Educator

Problem 85

Finding an Equation of a Hyperbola 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 .

Regina Hays

Numerade Educator

Problem 86

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

Matthew Lee

Numerade Educator

Problem 87

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 $(186,000 \text { miles per second). The difference }$ 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).

Regina Hays

Numerade Educator

Problem 88

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 $\left(x^{2} / 36\right)-\left(y^{2} / 64\right)=1 .$ At which point on the mirror will light from the point $(0,10)$ be reflected to the other focus?

KH

Kevin Hankins

Numerade Educator

Problem 89

Tangent Line Show that the equation of the tangent line to $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ at the point $\left(x_{0}, y_{0}\right)$ is $\left(\frac{x_{0}}{a^{2}}\right) x-\left(\frac{y_{0}}{b^{2}}\right) y=1$

Regina Hays

Numerade Educator

Problem 90

Proof Prove that the graph of the equation
$$A x^{2}+C y^{2}+D x+E y+F=0$$
is one of the following (except in degenerate cases).
$$
\begin{array}{ll}{\text { Conic }} & {\text { Condition }} \\ {\text { (a) Circle }} & {A=C} \\ {\text { (b) Parabola }} & {A=0 \text { or } C=0 \text { (but not both) }} \\ {\text { (c) Ellipse }} & {A C>0} \\ {\text { (d) Hyperbola }} & {A C<0}\end{array}
$$

Matthew Lee

Numerade Educator

Problem 91

True or False? In Exercises $91-96$ , 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.

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 92

True or False? In Exercises $91-96$ , 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.

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 93

True or False? In Exercises $91-96$ , 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
$$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, \quad b < a$$
then $2 \pi b \leq C \leq 2 \pi a$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 94

True or False? In Exercises $91-96$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $D \neq 0$ or $E \neq 0,$ then the graph of $y^{2}-x^{2}+D x+E y=0$ is a hyperbola.

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 95

True or False? In Exercises $91-96$ , 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 $\left(x^{2} / a^{2}\right)-\left(y^{2} / b^{2}\right)=1$ intersect at right angles, then $a=b$ .

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 96

True or False? In Exercises $91-96$ , 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.

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 97

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 $\left(P F_{1}\right)\left(P F_{2}\right) d^{2}$ is constant as $P$ varies on the ellipse, where $P F_{1}$ and $P F_{2}$ are the distances from $P$ to the foci $F_{1}$ and $F_{2}$ of the ellipse.

Regina Hays

Numerade Educator

Problem 98

Find the minimum value of
$$
(u-v)^{2}+\left(\sqrt{2-u^{2}}-\frac{9}{v}\right)^{2}
$$
for $0 < u < \sqrt{2}$ and $v > 0$

Matthew Lee

Numerade Educator