Calculus: One and Several Variables

Saturnino L. Salas, Garret J. Etgen, Einar Hille

Chapter 10

The Conic Sections; Polar Coordinates; Parametric Equations - all with Video Answers

Educators

Section 1

Geometry of Parabola. Ellipse, Hyperbola

Problem 1

Find the vertex, focus, axis, and directrix of the given parabola. Then sketch the parabola.
$$y=\frac{1}{2} x^{2}$$

Tyler Moulton

Tyler Moulton

Numerade Educator

Problem 2

Find the vertex, focus, axis, and directrix of the given parabola. Then sketch the parabola.
$$y=-\frac{1}{2} x^{2}$$

Tyler Moulton

Numerade Educator

Problem 3

Find the vertex, focus, axis, and directrix of the given parabola. Then sketch the parabola.
$$y=\frac{1}{2}(x-1)^{2}$$

Tyler Moulton

Numerade Educator

Problem 4

Find the vertex, focus, axis, and directrix of the given parabola. Then sketch the parabola.
$$y=-\frac{1}{2}(x-1)^{2}$$

Tyler Moulton

Numerade Educator

Problem 5

Find the vertex, focus, axis, and directrix of the given parabola. Then sketch the parabola.
$$y+2=\frac{1}{4}(x-2)^{2}$$

Tyler Moulton

Numerade Educator

Problem 6

Find the vertex, focus, axis, and directrix of the given parabola. Then sketch the parabola.
$$y-2=\frac{1}{4}(x+2)^{2}$$

Tyler Moulton

Numerade Educator

Problem 7

Find the vertex, focus, axis, and directrix of the given parabola. Then sketch the parabola.
$$y-x^{2}-4 x$$

Tyler Moulton

Numerade Educator

Problem 8

Find the vertex, focus, axis, and directrix of the given parabola. Then sketch the parabola.
$$y=x^{2}+x+1$$

Tyler Moulton

Numerade Educator

Problem 9

An ellipse is given. Find the center, the foci, the length of the major axis, and the length of the minor axis. Then sketch the ellipse.
$$x^{2} / 9+y^{2} / 4=1$$

Tyler Moulton

Numerade Educator

Problem 10

An ellipse is given. Find the center, the foci, the length of the major axis, and the length of the minor axis. Then sketch the ellipse.
$$x^{2} / 4+y / 9-1$$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 11

An ellipse is given. Find the center, the foci, the length of the major axis, and the length of the minor axis. Then sketch the ellipse.
$$3 x^{2}+2 y^{2}=12$$

Tyler Moulton

Numerade Educator

Problem 12

An ellipse is given. Find the center, the foci, the length of the major axis, and the length of the minor axis. Then sketch the ellipse.
$$3 x^{2}+4 y^{2}-12=0$$

Tyler Moulton

Numerade Educator

Problem 13

An ellipse is given. Find the center, the foci, the length of the major axis, and the length of the minor axis. Then sketch the ellipse.
$$4 x^{2}+9 y^{2}-18 y=27$$

Tyler Moulton

Numerade Educator

Problem 14

An ellipse is given. Find the center, the foci, the length of the major axis, and the length of the minor axis. Then sketch the ellipse.
$$4 x^{2}+y^{2}-6 y+5=0$$

Tyler Moulton

Numerade Educator

Problem 15

An ellipse is given. Find the center, the foci, the length of the major axis, and the length of the minor axis. Then sketch the ellipse.
$$4(x-1)^{2}+y^{2}=64$$

Tyler Moulton

Numerade Educator

Problem 16

An ellipse is given. Find the center, the foci, the length of the major axis, and the length of the minor axis. Then sketch the ellipse.
$$16(x-2)^{2}+25(y-3)^{2}=400$$

Tyler Moulton

Numerade Educator

Problem 17

A hyperbola is given. Find the center, the vertices, the foci, the asymptotes, and the length of the transverse axis. Then sketch the hyperbola.
$$x^{2}-y^{2}=1$$

Tyler Moulton

Numerade Educator

Problem 18

A hyperbola is given. Find the center, the vertices, the foci, the asymptotes, and the length of the transverse axis. Then sketch the hyperbola.
$$y^{2}-x^{2}=1$$

Tyler Moulton

Numerade Educator

Problem 19

A hyperbola is given. Find the center, the vertices, the foci, the asymptotes, and the length of the transverse axis. Then sketch the hyperbola.
$$x^{2} / 9-y^{2} / 16=1$$

Tyler Moulton

Numerade Educator

Problem 20

A hyperbola is given. Find the center, the vertices, the foci, the asymptotes, and the length of the transverse axis. Then sketch the hyperbola.
$$x^{2} / 16-y^{2} / 9=1$$

Tyler Moulton

Numerade Educator

Problem 21

A hyperbola is given. Find the center, the vertices, the foci, the asymptotes, and the length of the transverse axis. Then sketch the hyperbola.
$$y^{2} / 16-x^{2} / 9=1$$

Tyler Moulton

Numerade Educator

Problem 22

A hyperbola is given. Find the center, the vertices, the foci, the asymptotes, and the length of the transverse axis. Then sketch the hyperbola.
$$y^{2} / 9-x^{2} / 16=1$$

Tyler Moulton

Numerade Educator

Problem 23

A hyperbola is given. Find the center, the vertices, the foci, the asymptotes, and the length of the transverse axis. Then sketch the hyperbola.
$$(x-5)^{2} / 9-(y-3)^{2} / 16=1$$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 24

A hyperbola is given. Find the center, the vertices, the foci, the asymptotes, and the length of the transverse axis. Then sketch the hyperbola.
$$(x-1)^{2} / 16-(v-3)^{2} / 9=1$$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 25

A hyperbola is given. Find the center, the vertices, the foci, the asymptotes, and the length of the transverse axis. Then sketch the hyperbola.
$$4 x^{2}-8 x-y^{2}-6 y-1=0$$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 26

A hyperbola is given. Find the center, the vertices, the foci, the asymptotes, and the length of the transverse axis. Then sketch the hyperbola.
$$-3 x^{2}+3^{2}-6 x=0$$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 27

A parabola intersect a rectangle of area $A$ at two opposite vertices. Show that, if one side of the rectangle falls on the axis of the parabola, then the parabola subdivides the rectangle into two pieces, one of area $\frac{1}{3} A,$ the other of area $\frac{2}{3} A.$

Pawan Yadav

Numerade Educator

Problem 28

A line through the focus of a parabola intersects the parabola at two points $P$ and $Q$. Show that the tangent line through $P$ is perpendicular to the tangent line through $Q$.

Carson Merrill

Numerade Educator

Problem 29

Show that the graph of every quadratic function $y=a x^{2}+b x+c$ is a parabola . Find the vertex $_{1}$ the focus, the axis, and the directrix.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 30

Find the centroid of the first-quadrant portion of the elliptical region $b^{2} x^{2}+a^{2} y^{2} \leq a^{2} b^{2}.$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 31

Find the center, the vertices, the foci, the asymptotes, and the length of the transverse axis of the hyperbola with equation $x y=1 .$ HTNT: Define new $X Y$ -coordinates by selling $x=X+Y$ and $y=X-Y.$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 32

As $t$ ranges from 0 to $2 x,$ the points $(a \cos t, b \sin t)$ generate a curve in the $x y$ -plane. Identify the curve.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 33

An ellipse has area $A$ and major axis of length $2 a$. What is the distance between the foci?

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 34

A searchlight reflector is in the shape of a parabolic mirror. If it is 5 feet in diameter and 2 feet deep at the center, how far is the focus from the vertex of the mirror?

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 35

The line that passes through the focus of a parabola and is parallel to the directrix intersects the parabola at two points $A$ and $B$. The line segment $\overline{A B}$ is called the latus rectum of the parabola. In Exercises $35-38$ we work with the parabola $x^{2}=4 c y, c>0$ By $\Omega$ we mean the region bounded below by the parabola and above by the latus rectum.
Find the length of the latus rectum.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 36

What is the slope of the parabola at the endpoints of the latus rectum?

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 37

Determine the area of $\Omega$ and locate the centroid.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 38

Find the volume of the solid generated by revolving $\Omega$ about the $y$ -axis and locate the centroid of the solid. (For the centroid formulas, see Project $6.4 .)$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 39

Suppose that is flexible inclastic cable (see the figure) fixed at the ends supports a horizontal load. (lmagine a suspension bridge and think of the load on the cable as the roadivay.) Show that, if the load his constant weight per unit length, then the cable hangs in the form of a parabola.
HINT: The part of the cable that supports the load from 0 to $x$ is subject to the following forces:

RG

Ryan Griffith

Numerade Educator

Problem 40

(1) the weight of the load, which in this case is proportional
(2) the horizontal pull at $0: p(0)$
(3) the tangential pull at $x: \rho(x)$ Balancing the vertical forces, we have
$$
k x=p(x) \sin \theta . \quad \text { (weight - vertical pull at } x \text { ) }
$$
Balancing the horizontal forces, we have
$$
p(0)=p(x) \cos \theta
$$
(pull at 0 . horizontal pull al $x$ )
A lighting panel is perpendicular to the axis of a parabolic mirror. Show that all light rays beamed parallel to this axis are reflected to the focus of the mirror in paths of the same length.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 41

All equilateral triangles are similar; they differ only in scale. Show that the same is true of all parabolas.

Jay Patel

Numerade Educator

Problem 42

We refer to a hyperbola is standard position.
Find functions $x=x(t), y=y(t)$ such that, as $t$ ranges over the set of real numbers, the points $(x(t), y(t))$ traverse
(a) the right branch of the hyperbola.
(b) the left branch of the hyperbola.

Matthew Lee

Numerade Educator

Problem 43

We refer to a hyperbola is standard position.
Find the area of the region between the right branch of the hyperbola and the vertical line $x=2 a.$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 44

We refer to a hyperbola is standard position.
Show that at each point $P$ of the hyperbola the tangent line bisects the angle between the focal radii $\overline{F_{1} P}$ and $\overline{F_{2} P}.$
Although all parabolas have exactly the same shape (Exercise
41), ellipses come in different shapes. The shape of an ellipse depends on its eccentricity e. This is half the distance between the foci divided by half the length of the major axis:
$$e=c / a$$
For every ellipse, $0<e<1$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 45

Determine the eccentricity of the ellipse.
$$x^{2} / 25+y^{2} / 16=1$$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 46

Determine the eccentricity of the ellipse.
$$x^{2} / 16+y^{2} / 25=1$$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 47

Determine the eccentricity of the ellipse.
$$(x-1)^{2} / 25+(y+2)^{2} / 9=1$$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 48

Determine the eccentricity of the ellipse.
$$(x+1)^{2} / 169+(y-1)^{2} / 144=1$$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 49

Suppose that $E_{1}$ and $E_{2}$ are both ellipses with the same major axis. Compare the shape of $E_{1}$ to the shape of $E_{2}$ if $e_{1}-c_{2},$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 50

What happens to an ellipse with major axis $2 a$ if $e$ tends to $0 ?$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 51

What happens to an ellipse with major axis $2 a$ if $e$ tends to $1 ?$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 52

Write an equation for the ellipse.
Major axis from (-3,0) to $(3,0),$ eccentricity $_{3}^{1}.$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 53

Write an equation for the ellipse.
Major axis from (-3.0) to $(3,0),$ eccentricity $_{3}^{2} \sqrt{2}.$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 54

Let $l$ be a line and let $F$ be a point not on $l .$ You have seen that the set of points $P$ for which
$$
d(F, P)=d(l, P)
$$
is a parabola. Show that, if $0<e<1$, then the set of all points $P$ for which
$$
d(F, P)=e d(l, P)
$$
is an ellipse of eccentricity $e .$ HINT: Begin by choosing a coordinate system whereby $F$ falls on the origin and $/$ is a vertical line $x=k.$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 55

Determine the eccentricity of the hyperbola.
$$x^{2}/9-y^{2} / 16=1$$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 56

Determine the eccentricity of the hyperbola.
$$x^{2} / 16-y^{2} / 9=1$$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 57

Determine the eccentricity of the hyperbola.
$$x^{2}-y^{2}=1$$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 58

Determine the eccentricity of the hyperbola.
$$x^{2} / 25-y^{2} / 144=1$$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 59

Suppose $H_{1}$ and $H_{2}$ are both hyperbolas with the same transverse axis. Compare the shape of $H_{1}$ to the shape of $H_{2}$ if $e_{1}<c_{2}.$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 60

What happens to a hyperbola if $e$ tends to $1 ?$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 61

What happens to a hyperbola if $e$ increases without bound?

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 62

(Compare to Exercise $54 .$ ) Let $/$ be a line and let $F$ be a point not on $l .$ Show that, if $e>1,$ then the set of all points $P$ for which
$$
d(F, P)=e d(l, P)
$$
is a hyperbola of eccentricity, e. HINT: Begin by choosing a coordinate system whereby $F$ falls on the origin and $l$ is a vertical line $x=k.$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 63

Show that every parabola has an equation of the form
$$
(\alpha x+\beta y)^{2}=\gamma x+\delta y+\epsilon \quad \text { with } \quad \alpha^{2}=\beta^{2} \leftrightarrow 0
$$
HINT: Take $l: A x+B y+C=0$ as the directrix, $F(a, b)$ as the focus.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator