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Calculus of a Single Variable

Ron Larson

Chapter 10

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

Educators

+ 5 more educators

Section 1

Conics and Calculus

06:32

Problem 1

Conic Sections State the definitions of parabola, ellipse, and hyperbola in your own words.

Shuyang Fu
Shuyang Fu
Numerade Educator
01:04

Problem 2

Reflective Property Use a sketch to illustrate the reflective property of an ellipse.

Matthew Lee
Matthew Lee
Numerade Educator
03:37

Problem 3

Eccentricity Consider an ellipse with eccentricity $e$ .
(a) What are the possible values of $e ?$
(b) What happens to the graph of the ellipse as $e$ increases?

Shuyang Fu
Shuyang Fu
Numerade Educator
01:50

Problem 4

Hyperbola Explain how to sketch a hyperbola with a vertical transverse axis.

Matthew Lee
Matthew Lee
Numerade Educator
01:20

Problem 5

Matching In Exercises $5-10$ , match the equation with its graph. [The graphs are labeled (a), (b), ( ), (d), (e), and (f).]
$$y^{2}=4 x$$

Shuyang Fu
Shuyang Fu
Numerade Educator
01:49

Problem 6

Matching In Exercises $5-10$ , match the equation with its graph. [The graphs are labeled (a), (b), ( ), (d), (e), and (f).]
$$(x+4)^{2}=-2(y-2)$$

Matthew Lee
Matthew Lee
Numerade Educator
01:39

Problem 7

Matching In Exercises $5-10$ , match the equation with its graph. [The graphs are labeled (a), (b), ( ), (d), (e), and (f).]
$$\frac{y^{2}}{16}-\frac{x^{2}}{1}=1$$

Shuyang Fu
Shuyang Fu
Numerade Educator
01:43

Problem 8

Matching In Exercises $5-10$ , match the equation with its graph. [The graphs are labeled (a), (b), ( ), (d), (e), and (f).]
$$\frac{(x-2)^{2}}{16}+\frac{(y+1)^{2}}{4}=1$$

Matthew Lee
Matthew Lee
Numerade Educator
01:36

Problem 9

Matching In Exercises $5-10$ , match the equation with its graph. [The graphs are labeled (a), (b), ( ), (d), (e), and (f).]
$$\frac{x^{2}}{4}+\frac{y^{2}}{9}=1$$

Shuyang Fu
Shuyang Fu
Numerade Educator
02:01

Problem 10

Matching In Exercises $5-10$ , match the equation with its graph. [The graphs are labeled (a), (b), ( ), (d), (e), and (f).]
$$\frac{(x-2)^{2}}{9}-\frac{y^{2}}{4}=1$$

Matthew Lee
Matthew Lee
Numerade Educator
05:50

Problem 11

Sketching a Parabola In Exercises $11-16,$ find the vertex, focus, and directrix of the parabola, and sketch its graph.
$$(x+5)+(y-3)^{2}=0$$

Shuyang Fu
Shuyang Fu
Numerade Educator
03:44

Problem 12

Sketching a Parabola In Exercises $11-16,$ find the vertex, focus, and directrix of the parabola, and sketch its graph.
$$(x-6)^{2}-2(y+7)=0$$

Matthew Lee
Matthew Lee
Numerade Educator
12:59

Problem 13

Sketching a Parabola In Exercises $11-16,$ find the vertex, focus, and directrix of the parabola, and sketch its graph.
$$y^{2}-4 y-4 x=0$$

GC
Georgy Chargaziya
Numerade Educator
05:17

Problem 14

Sketching a Parabola In Exercises $11-16,$ find the vertex, focus, and directrix of the parabola, and sketch its graph.
$$y^{2}+6 y+8 x+25=0$$

Matthew Lee
Matthew Lee
Numerade Educator
06:09

Problem 15

Sketching a Parabola In Exercises $11-16,$ find the vertex, focus, and directrix of the parabola, and sketch its graph.
$$x^{2}+4 x+4 y-4=0$$

DS
Daniel Stasiuk
Numerade Educator
04:09

Problem 16

Sketching a Parabola In Exercises $11-16,$ find the vertex, focus, and directrix of the parabola, and sketch its graph.
$$x^{2}-2 x-4 y-7=0$$

Matthew Lee
Matthew Lee
Numerade Educator
03:07

Problem 17

Finding the Standard Equation of a Parabola In Exercises $17-24$ , find the standard form of the equation of the parabola with the given characteristics.
Vertex: $(5,4)$
Focus: $(3,4)$

Shuyang Fu
Shuyang Fu
Numerade Educator
01:54

Problem 18

Finding the Standard Equation of a Parabola In Exercises $17-24$ , find the standard form of the equation of the parabola with the given characteristics.
Vertex: $(-3,-1)$
Focus: $(-3,1)$

Matthew Lee
Matthew Lee
Numerade Educator
02:30

Problem 19

Finding the Standard Equation of a Parabola In Exercises $17-24$ , find the standard form of the equation of the parabola with the given characteristics.
Vertex: $(0,5)$
Directrix: $y=-3$

Shuyang Fu
Shuyang Fu
Numerade Educator
02:44

Problem 20

Finding the Standard Equation of a Parabola In Exercises $17-24$ , find the standard form of the equation of the parabola with the given characteristics.
Focus: $(2,2)$
Directrix: $x=-2$

Matthew Lee
Matthew Lee
Numerade Educator
02:57

Problem 21

Finding the Standard Equation of a Parabola In Exercises $17-24$ , find the standard form of the equation of the parabola with the given characteristics.
Vertex: $(1,-1)$
Points on the parabola:
$(-1,-4),(3,-4)$

Shuyang Fu
Shuyang Fu
Numerade Educator
03:04

Problem 22

Finding the Standard Equation of a Parabola In Exercises $17-24$ , find the standard form of the equation of the parabola with the given characteristics.
Vertex: $(2,4)$
Points on the parabola:
$(0,0),(4,0)$

Matthew Lee
Matthew Lee
Numerade Educator
06:50

Problem 23

Finding the Standard Equation of a Parabola In Exercises $17-24$ , find the standard form of the equation of the parabola with the given characteristics.
Axis is parallel to $y$ -axis; graph passes through $(0,3),(3,4),$ and $(4,11) .$

Shuyang Fu
Shuyang Fu
Numerade Educator
03:48

Problem 24

Finding the Standard Equation of a Parabola In Exercises $17-24$ , find the standard form of the equation of the parabola with the given characteristics.
Directrix: $y=-2 ;$ endpoints of latus rectum are $(0,2)$ and $(8,2) .$

Matthew Lee
Matthew Lee
Numerade Educator
03:59

Problem 25

Sketching an Ellipse In Exercises $25-30$ , find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph.
$$16 x^{2}+y^{2}=16$$

Shuyang Fu
Shuyang Fu
Numerade Educator
05:20

Problem 26

Sketching an Ellipse In Exercises $25-30$ , find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph.
$$3 x^{2}+7 y^{2}=63$$

Matthew Lee
Matthew Lee
Numerade Educator
05:32

Problem 27

Sketching an Ellipse In Exercises $25-30$ , find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph.
$$\frac{(x-3)^{2}}{16}+\frac{(y-1)^{2}}{25}=1$$

KA
Karim Arabi
Numerade Educator
05:04

Problem 28

Sketching an Ellipse In Exercises $25-30$ , find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph.
$$(x+4)^{2}+\frac{(y+6)^{2}}{1 / 4}=1$$

Matthew Lee
Matthew Lee
Numerade Educator
07:45

Problem 29

Sketching an Ellipse In Exercises $25-30$ , find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph.
$$9 x^{2}+4 y^{2}+36 x-24 y-36=0$$

Shuyang Fu
Shuyang Fu
Numerade Educator
08:35

Problem 30

Sketching an Ellipse In Exercises $25-30$ , find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph.
$$x^{2}+10 y^{2}-6 x+20 y+18=0$$

Matthew Lee
Matthew Lee
Numerade Educator
02:38

Problem 31

Finding the Standard Equation of an Ellipse In Exercises $31-36,$ find the standard form of the equation of the ellipse with the given characteristics.
Center: $(0,0)$
Focus: $(5,0)$
Vertex: $(6,0)$

Regina Hays
Regina Hays
Numerade Educator
03:47

Problem 32

Finding the Standard Equation of an Ellipse In Exercises $31-36,$ find the standard form of the equation of the ellipse with the given characteristics.
Vertices: $(0,3),(8,3)$
Eccentricity: $\frac{3}{4}$

Matthew Lee
Matthew Lee
Numerade Educator
03:23

Problem 33

Finding the Standard Equation of an Ellipse In Exercises $31-36,$ find the standard form of the equation of the ellipse with the given characteristics.
Vertices: $(3,1),(3,9)$
Minor axis length: 6

Shuyang Fu
Shuyang Fu
Numerade Educator
02:59

Problem 34

Finding the Standard Equation of an Ellipse In Exercises $31-36,$ find the standard form of the equation of the ellipse with the given characteristics.
Foci: $(0, \pm 9)$
Major axis length: 22

Matthew Lee
Matthew Lee
Numerade Educator
03:04

Problem 35

Finding the Standard Equation of an Ellipse In Exercises $31-36,$ find the standard form of the equation of the ellipse with the given characteristics.
Center: $(0,0)$
Major axis: horizontal
Points on the ellipse:
$(3,1),(4,0)$

Shuyang Fu
Shuyang Fu
Numerade Educator
03:54

Problem 36

Finding the Standard Equation of an Ellipse In Exercises $31-36,$ find the standard form of the equation of the ellipse with the given characteristics.
Center: $(1,2)$
Major axis: vertical
Points on the ellipse:
$(1,6),(3,2)$

Matthew Lee
Matthew Lee
Numerade Educator
09:14

Problem 37

Sketching a Hyperbola In Exercises $37-40$ , find the center, foci, vertices, and eccentricity of the hyperbola, and sketch its graph using asymptotes as an aid.
$$\frac{x^{2}}{25}-\frac{y^{2}}{16}=1$$

KK
Keegan Kirk
Numerade Educator
06:49

Problem 38

Sketching a Hyperbola In Exercises $37-40$ , find the center, foci, vertices, and eccentricity of the hyperbola, and sketch its graph using asymptotes as an aid.
$$\frac{(y+3)^{2}}{225}-\frac{(x-5)^{2}}{64}=1$$

Matthew Lee
Matthew Lee
Numerade Educator
09:39

Problem 39

Sketching a Hyperbola In Exercises $37-40$ , find the center, foci, vertices, and eccentricity of the hyperbola, and sketch its graph using asymptotes as an aid.
$$9 x^{2}-y^{2}-36 x-6 y+18=0$$

Shuyang Fu
Shuyang Fu
Numerade Educator
07:27

Problem 40

Sketching a Hyperbola In Exercises $37-40$ , find the center, foci, vertices, and eccentricity of the hyperbola, and sketch its graph using asymptotes as an aid.
$$y^{2}-16 x^{2}+64 x-208=0$$

Matthew Lee
Matthew Lee
Numerade Educator
03:31

Problem 41

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

Regina Hays
Regina Hays
Numerade Educator
04:41

Problem 42

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

Matthew Lee
Matthew Lee
Numerade Educator
04:56

Problem 43

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

Shuyang Fu
Shuyang Fu
Numerade Educator
03:07

Problem 44

Finding the Standard Equation of a a Hyperbola In Exercises $41-48,$ find the standard form of the equation of the hyperbola with the given characteristics.
Vertices: $(2, \pm 3)$
Foci: $(2, \pm 5)$

Matthew Lee
Matthew Lee
Numerade Educator
02:34

Problem 45

Finding the Standard Equation of a a Hyperbola In Exercises $41-48,$ find the standard form of the equation of the hyperbola with the given characteristics.
Center: $(0,0)$
Vertex: $(0,2)$
Focus: $(0,4)$

Shuyang Fu
Shuyang Fu
Numerade Educator
04:34

Problem 46

Finding the Standard Equation of a a Hyperbola In Exercises $41-48,$ find the standard form of the equation of the hyperbola with the given characteristics.
$\begin{aligned} \text { Vertices: }(0,2), &(6,2) \\ \text { Asymptotes: } y &=\frac{2}{3} x \\ y &=4-\frac{2}{3} x \end{aligned}$

Matthew Lee
Matthew Lee
Numerade Educator
03:42

Problem 47

Finding the Standard Equation of a Hyperbola In Exercises $41-48,$ find the standard form of the equation of the hyperbola with the given characteristics.
$$\begin{aligned} \text { Vertices: }(0,2) &,(6,2) \\ \text { Asyruplotes: } y &=\frac{2}{7} x \\ y &=4-\frac{2}{3} x \end{aligned}$$

Shuyang Fu
Shuyang Fu
Numerade Educator
07:19

Problem 48

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

Matthew Lee
Matthew Lee
Numerade Educator
09:36

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 $x .$ The normal lines to point is perpendicular to the tangent line at the point.)
$$\frac{x^{2}}{9}-y^{2}=1, \quad x=6$$

Regina Hays
Regina Hays
Numerade Educator
07:51

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 $x .$ The normal lines to point is perpendicular to the tangent line at the point.)
$$\frac{y^{2}}{4}-\frac{x^{2}}{2}=1, \quad x=4$$

Matthew Lee
Matthew Lee
Numerade Educator
06:45

Problem 51

Classifying the Graph of an Equation In Exercises $51-56$ , classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.
$$25 x^{2}-10 x-200 y-119=0$$

AR
Amanda Reeder
Numerade Educator
01:28

Problem 52

Classifying the Graph of an Equation In Exercises $51-56$ , classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.
$$4 x^{2}-y^{2}-4 x-3=0$$

Matthew Lee
Matthew Lee
Numerade Educator
05:12

Problem 53

Classifying the Graph of an Equation In Exercises $51-56$ , 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
Brian Austin
Numerade Educator
02:13

Problem 54

Classifying the Graph of an Equation In Exercises $51-56$ , 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
Matthew Lee
Numerade Educator
04:09

Problem 55

Classifying the Graph of an Equation In Exercises $51-56$ , classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.
$$9 x^{2}+9 y^{2}-36 x+6 y+34=0$$

Shuyang Fu
Shuyang Fu
Numerade Educator
00:33

Problem 56

Classifying the Graph of an Equation In Exercises $51-56$ , classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.
$$y^{2}-4 y=x+5$$

Matthew Lee
Matthew Lee
Numerade Educator
03:50

Problem 57

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) Describe one way you could change the original equation so that its graph is a parabola.

Regina Hays
Regina Hays
Numerade Educator
00:55

Problem 58

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.

Matthew Lee
Matthew Lee
Numerade Educator
04:01

Problem 59

Ellipse Let $C$ be the circumference of the ellipse $\left(x^{2} / a^{2}\right)+\left(y^{2} / b^{2}\right)=1, \quad b<a .$ Explain why $2 \pi b<C<2 \pi a .$ Use a graph to support your explanation.Ellipse Let $C$ be the circumference of the ellipse $\left(x^{2} / a^{2}\right)+\left(y^{2} / b^{2}\right)=1, \quad b<a .$ Explain why $2 \pi b<C<2 \pi a .$ Use a graph to support your explanation.

Regina Hays
Regina Hays
Numerade Educator
08:04

Problem 60

HOW DO YOU SEE IT? Describe in words how a plane could intersect with the double-napped cone to form each conic section (see figure).

Regina Hays
Regina Hays
Numerade Educator
04:23

Problem 60

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

Matthew Lee
Matthew Lee
Numerade Educator
04:42

Problem 61

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
Regina Hays
Numerade Educator
02:20

Problem 62

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
Matthew Lee
Numerade Educator
05:35

Problem 63

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
Regina Hays
Numerade Educator
16:55

Problem 64

Proof
(a) Prove that if any two tangent lines to a parabola intersect at right angles, then 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 their point of intersection lies on the directrix.

Regina Hays
Regina Hays
Numerade Educator
02:18

Problem 65

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. Find an equation for the parabolic shape of the cable.

Regina Hays
Regina Hays
Numerade Educator
04:34

Problem 66

Arc Length Find the length of the parabolic cable in Exercise $65 .$

Regina Hays
Regina Hays
Numerade Educator
11:06

Problem 67

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

Regina Hays
Regina Hays
Numerade Educator
05:47

Problem 68

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
$2 \pi \int_{0}^{r} x \sqrt{1+\left(\frac{x}{10}\right)^{2}} d x=\frac{\pi {15}\left[\left(100+r^{2}\right)^{3 / 2}-1000\right]$

Regina Hays
Regina Hays
Numerade Educator
06:02

Problem 69

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
Regina Hays
Numerade Educator
03:19

Problem 70

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

Matthew Lee
Matthew Lee
Numerade Educator
03:47

Problem 71

Explorer 1 On January $31,1958,$ the United States launched the research satellite Explorer $1 .$ Its low and high points above the surface of Earth were 220 miles and 1563 miles. Find the eccentricity of its elliptical orbit. (Use 4000 miles as the radius of Earth.)

Regina Hays
Regina Hays
Numerade Educator
03:46

Problem 72

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. (Use 4000 miles as the radius of Earth.)

Matthew Lee
Matthew Lee
Numerade Educator
02:00

Problem 73

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 $\mathrm{A} \mathrm{U}$ ( astronomical unit is approximately $92.956 \times 10^{6}$ miles $)$ and its minimum distance is approximately 0.59 AU. Find the eccentricity of the orbit.

Regina Hays
Regina Hays
Numerade Educator
03:11

Problem 74

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
Matthew Lee
Numerade Educator
28:18

Problem 75

Area, Volume, and Surface Area In Exercises 75 and 76 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).
$$\frac{x^{2}}{4}+\frac{y^{2}}{1}=1$$

Regina Hays
Regina Hays
Numerade Educator
23:18

Problem 76

Area, Volume, and Surface Area In Exercises 75 and 76 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
Regina Hays
Numerade Educator
05:46

Problem 77

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
Regina Hays
Numerade Educator
04:30

Problem 78

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
Matthew Lee
Numerade Educator
03:03

Problem 79

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

Regina Hays
Regina Hays
Numerade Educator
17:39

Problem 80

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). IHint: (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 $\theta$ between two lines with slopes $m_{1}$ and $m_{2}$ ,
$\tan \theta=\left|\frac{m_{1}-m_{2}}{1+m_{1} m_{2}}\right| ]$

Regina Hays
Regina Hays
Numerade Educator
04:35

Problem 81

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
Regina Hays
Numerade Educator
13:21

Problem 82

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
Matthew Lee
Numerade Educator
06:24

Problem 83

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 when the time difference between the pulses from the transmitting stations is 1000 microseconds $(0.001$ second $)$ .

Regina Hays
Regina Hays
Numerade Educator
05:47

Problem 84

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

Regina Hays
Regina Hays
Numerade Educator
05:35

Problem 85

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
Regina Hays
Numerade Educator
16:29

Problem 86

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
Matthew Lee
Numerade Educator
01:49

Problem 87

True or False? In Exercises $87-92,$ 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.

Willis James
Willis James
Numerade Educator
00:59

Problem 88

The point on a parabola closest to its focus is its vertex.

Matthew Lee
Matthew Lee
Numerade Educator
03:20

Problem 89

The eccentricity of a hyperbola with a horizontal transverse axis is $e=\sqrt{1+m^{2}},$ where $m$ and $-m$ the slopes of the asymptotes.

Regina Hays
Regina Hays
Numerade Educator
01:30

Problem 90

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

Matthew Lee
Matthew Lee
Numerade Educator
02:47

Problem 91

If the asymptotes of the hyperbola $\left(x^{2} / a^{2}\right)-\left(y^{2} / b^{2}\right)=1$ intersect at right angles, then $a=b$ .

Regina Hays
Regina Hays
Numerade Educator
01:33

Problem 92

Every tangent line to a hyperbola intersects the hyperbola only at the point of tangency.

Matthew Lee
Matthew Lee
Numerade Educator
20:46

Problem 93

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
Regina Hays
Numerade Educator
08:54

Problem 94

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
Matthew Lee
Numerade Educator