Modern Analytic Geometry

William Wooton, Edwin F. Beckenbach, Frank J. Fleming

Chapter 5

Transformation of Coordinates - all with Video Answers

Educators

Section 1

Translation of Axes

Problem 1

an $x y$-equation and the $x y$-coordinates of a point $S$ are given. Find an $x^{\prime} y^{\prime}$-equation for the graph of the given equation if the origin of the $x^{\prime} y^{\prime}$-system is at the point $\mathbf{S}$.
$x-4 y^2+16 y-7=0 ; \mathbf{S}(-3,4)$

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Problem 2

an $x y$-equation and the $x y$-coordinates of a point $S$ are given. Find an $x^{\prime} y^{\prime}$-equation for the graph of the given equation if the origin of the $x^{\prime} y^{\prime}$-system is at the point $\mathbf{S}$.
$x^2-y^2-6 x+10 y-20=0 ; \mathbf{S}(3,5)$

Caleb Fink

Numerade Educator

Problem 3

an $x y$-equation and the $x y$-coordinates of a point $S$ are given. Find an $x^{\prime} y^{\prime}$-equation for the graph of the given equation if the origin of the $x^{\prime} y^{\prime}$-system is at the point $\mathbf{S}$.
$x^2+4 y^2-6 x-16 y-11=0 ; \mathbf{S}(3,2)$

Samuel Coplin

Numerade Educator

Problem 4

an $x y$-equation and the $x y$-coordinates of a point $S$ are given. Find an $x^{\prime} y^{\prime}$-equation for the graph of the given equation if the origin of the $x^{\prime} y^{\prime}$-system is at the point $\mathbf{S}$.
$x^2+y^2-8 x+10 y-4=0 ; \mathbf{S}(4,-5)$

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Problem 5

an $x y$-equation and the $x y$-coordinates of a point $S$ are given. Find an $x^{\prime} y^{\prime}$-equation for the graph of the given equation if the origin of the $x^{\prime} y^{\prime}$-system is at the point $\mathbf{S}$.
$x^2+2 y^2-2 x+16 y+33=0 ; \mathbf{S}(1,-4)$

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Problem 6

an $x y$-equation and the $x y$-coordinates of a point $S$ are given. Find an $x^{\prime} y^{\prime}$-equation for the graph of the given equation if the origin of the $x^{\prime} y^{\prime}$-system is at the point $\mathbf{S}$.
$4 x^2-y^2-12 x-6 y+24=0 ; \mathbf{S}\left(\frac{3}{2},-3\right)$

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Problem 7

find a Cartesian equation for the conic section satisfying the given conditions. Sketch the curve.
Parabola with vertex $\mathbf{V}(3,-2)$ and focus $\mathbf{F}(3,4)$.

Wendi Zhao

Numerade Educator

Problem 8

find a Cartesian equation for the conic section satisfying the given conditions. Sketch the curve.
Parabola with vertex $\mathbf{V}\left(-\frac{3}{2}, 2\right)$ and focus $\mathbf{F}(1,2)$.

Sunni Burns

Numerade Educator

Problem 9

find a Cartesian equation for the conic section satisfying the given conditions. Sketch the curve.
Parabola with focus $\mathbf{F}(2,4)$ and directrix the line with equation $x=8$.

Kacie Vlach

Numerade Educator

Problem 10

find a Cartesian equation for the conic section satisfying the given conditions. Sketch the curve.
Parabola with focus $\mathbf{F}(-1,2)$ and directrix the line with equation $y=-4$.

AG

Ankit Gupta

Numerade Educator

Problem 11

find a Cartesian equation for the conic section satisfying the given conditions. Sketch the curve.
Ellipse with foci $F_1(1,2)$ and $F_2(9,2)$, and major axis of length 10 .

Teresa Fuston

Numerade Educator

Problem 12

find a Cartesian equation for the conic section satisfying the given conditions. Sketch the curve.
Ellipse with foci $\mathbf{F}_1(-3,2)$ and $\mathbf{F}_2^{\prime}(-3,6)$, and major axis of length 12.

Teresa Fuston

Numerade Educator

Problem 13

find a Cartesian equation for the conic section satisfying the given conditions. Sketch the curve.
Ellipse with vertices $\mathbf{V}_1(8,2)$ and $\mathbf{V}_2(-4,2)$, and one focus $\mathbf{F}_1(6,2)$.

Kelly Brooks

Numerade Educator

Problem 14

find a Cartesian equation for the conic section satisfying the given conditions. Sketch the curve.
Ellipse with center $\mathbf{C}(-2,4)$, one vertex $\mathbf{V}_1(3,4)$, and associated focus $\mathrm{F}_1(2,4)$

AG

Ankit Gupta

Numerade Educator

Problem 15

find a Cartesian equation for the conic section satisfying the given conditions. Sketch the curve.
Hyperbola with foci $F_1(3,2)$ and $F_2(3,-6)$, and transverse axis of length 4 .

Wendi Zhao

Numerade Educator

Problem 16

find a Cartesian equation for the conic section satisfying the given conditions. Sketch the curve.
Hyperbola with foci $\mathbf{F}_1(-6,-3)$ and $\mathbf{F}_2(4,-3)$, and one vertex $\mathbf{V}_2(3,-3)$.

Wendi Zhao

Numerade Educator

Problem 17

find a Cartesian equation for the conic section satisfying the given conditions. Sketch the curve.
Hyperbola with vertices $\mathbf{V}_1(6,2)$ and $\mathbf{V}_2(-2,2)$, and eccentricity $\frac{7}{5}$.

Kelly Brooks

Numerade Educator

Problem 18

find a Cartesian equation for the conic section satisfying the given conditions. Sketch the curve.
Hyperbola with center $\mathbf{C}(3,2)$, one focus $\mathbf{F}_1(3,7)$, and eccentricity $\frac{5}{3}$.

Sheryl Ezze

Numerade Educator

Problem 19

find an $x^{\prime} y^{\prime}$-equation for the graph of the given $x y$-equation so that the center or, in the case of a parabola, the vertex, of the graph is at the origin in the $x^{\prime} y^{\prime}$-system. Sketch the curve, showing both sets of coordinate axes, and identify the type of curve.
$x^2+y^2+4 x-10 y-36=0$

Willis James

Numerade Educator

Problem 20

find an $x^{\prime} y^{\prime}$-equation for the graph of the given $x y$-equation so that the center or, in the case of a parabola, the vertex, of the graph is at the origin in the $x^{\prime} y^{\prime}$-system. Sketch the curve, showing both sets of coordinate axes, and identify the type of curve.
$x^2-4 y^2+4 x+32 y-64=0$

Willis James

Numerade Educator

Problem 21

find an $x^{\prime} y^{\prime}$-equation for the graph of the given $x y$-equation so that the center or, in the case of a parabola, the vertex, of the graph is at the origin in the $x^{\prime} y^{\prime}$-system. Sketch the curve, showing both sets of coordinate axes, and identify the type of curve.
$2 x^2+y^2+8 x-8 y-48=0$

Willis James

Numerade Educator

Problem 22

find an $x^{\prime} y^{\prime}$-equation for the graph of the given $x y$-equation so that the center or, in the case of a parabola, the vertex, of the graph is at the origin in the $x^{\prime} y^{\prime}$-system. Sketch the curve, showing both sets of coordinate axes, and identify the type of curve.
$x^2-4 x-4 y+16=0$

Sinisa Stura

Numerade Educator

Problem 23

find an $x^{\prime} y^{\prime}$-equation for the graph of the given $x y$-equation so that the center or, in the case of a parabola, the vertex, of the graph is at the origin in the $x^{\prime} y^{\prime}$-system. Sketch the curve, showing both sets of coordinate axes, and identify the type of curve.
$4 x+y^2+4 y-4=0$

Sinisa Stura

Numerade Educator

Problem 24

find an $x^{\prime} y^{\prime}$-equation for the graph of the given $x y$-equation so that the center or, in the case of a parabola, the vertex, of the graph is at the origin in the $x^{\prime} y^{\prime}$-system. Sketch the curve, showing both sets of coordinate axes, and identify the type of curve.
$4 x^2+9 y^2+8 x+36 y+4=0$

Cory Kuzinski

Numerade Educator

Problem 25

find an $x^{\prime} y^{\prime}$-equation for the graph of the given $x y$-equation so that the center or, in the case of a parabola, the vertex, of the graph is at the origin in the $x^{\prime} y^{\prime}$-system. Sketch the curve, showing both sets of coordinate axes, and identify the type of curve.
$4 x^2+4 y^2-12 x+8 y-3=0$

AG

Ankit Gupta

Numerade Educator

Problem 26

find an $x^{\prime} y^{\prime}$-equation for the graph of the given $x y$-equation so that the center or, in the case of a parabola, the vertex, of the graph is at the origin in the $x^{\prime} y^{\prime}$-system. Sketch the curve, showing both sets of coordinate axes, and identify the type of curve.
$8 x-y^2-8 y=0$

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Problem 27

find an $x^{\prime} y^{\prime}$-equation for the graph of the given $x y$-equation so that the center or, in the case of a parabola, the vertex, of the graph is at the origin in the $x^{\prime} y^{\prime}$-system. Sketch the curve, showing both sets of coordinate axes, and identify the type of curve.
$9 x^2+4 y^2-18 x+24 y+45=0$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 28

find an $x^{\prime} y^{\prime}$-equation for the graph of the given $x y$-equation so that the center or, in the case of a parabola, the vertex, of the graph is at the origin in the $x^{\prime} y^{\prime}$-system. Sketch the curve, showing both sets of coordinate axes, and identify the type of curve.
$x^2-y^2+6 x+4 y+5=0$

Caleb Fink

Numerade Educator

Problem 29

find an $x^{\prime} y^{\prime}$-equation for the graph of the given $x y$-equation so that the center or, in the case of a parabola, the vertex, of the graph is at the origin in the $x^{\prime} y^{\prime}$-system. Sketch the curve, showing both sets of coordinate axes, and identify the type of curve.
$4 x^2-9 y^2-16 x+18 y-7=0$

Km Neeraj

Numerade Educator

Problem 30

find an $x^{\prime} y^{\prime}$-equation for the graph of the given $x y$-equation so that the center or, in the case of a parabola, the vertex, of the graph is at the origin in the $x^{\prime} y^{\prime}$-system. Sketch the curve, showing both sets of coordinate axes, and identify the type of curve.
$4 y^2-3 x^2+8 y-12 x-16=0$

AG

Ankit Gupta

Numerade Educator

Problem 31

sketch the graph of the given equation.
$x^2+y^2-4 x+6 y+13=0$

AG

Ankit Gupta

Numerade Educator

Problem 32

sketch the graph of the given equation.
$4 x^2-9 y^2+16 x+18 y+7=0$

Tyler Moulton

Numerade Educator

Problem 33

sketch the graph of the given equation.
$x^2+2 y^2-4 y+3=0$

AG

Ankit Gupta

Numerade Educator

Problem 34

sketch the graph of the given equation.
$x^2-8 x+15=0$

Kathleen Carty

Numerade Educator

Problem 35

sketch the graph of the given equation.
$y^2+6 y+9=0$

AG

Ankit Gupta

Numerade Educator

Problem 36

sketch the graph of the given equation.
$x^2+10 x+30=0$

Erika Bustos

Numerade Educator

Problem 37

Use a translation of axes to remove first-degree terms from the equation $x y+4 x-8 y+6=0$; that is, find an $x^{\prime} y^{\prime}$-equation for the graph of the given $x y$-equation such that the first-degree terms in $x^{\prime}$ or $y^{\prime}$ have coefficient 0 . (Hint: Replace $x$ and $y$ with $x^{\prime}+h$ and $y^{\prime}+k$, respectively, and find suitable values for $h$ and $k$.)

Himanshu Kushwaha

Himanshu Kushwaha

Numerade Educator

Problem 38

Use the method suggested in Exercise 37 to remove first-degree terms from the equation

$$
x y+a x+b y+c=0 \text {. }
$$

Minh Le

Numerade Educator

Problem 39

refer to the equation $A x^2+C y^2+D x+E y+F=0$, where $A$ and $C$ are not both 0 .
Show that if $A=0, C \neq 0$, and $D \neq 0$, then the equation is equivalent to one of the form

$$
(y-k)^2=4 p(x-h)
$$

Carson Merrill

Numerade Educator

Problem 40

refer to the equation $A x^2+C y^2+D x+E y+F=0$, where $A$ and $C$ are not both 0 .
Show that if $A \neq 0, C=0$, and $E \neq 0$, then the equation is equivalent to one of the form

$$
(x-h)^2=4 p(y-k)
$$

Carson Merrill

Numerade Educator

Problem 41

refer to the equation $A x^2+C y^2+D x+E y+F=0$, where $A$ and $C$ are not both 0 .
Show that if $A=C \neq 0$, then the equation is equivalent to one of the form

$$
(x-h)^2+(y-k)^2=q .
$$

What must be true of $q$ if the equation is to have a nonempty graph in $\Omega^2$ ?

Carson Merrill

Numerade Educator

Problem 42

refer to the equation $A x^2+C y^2+D x+E y+F=0$, where $A$ and $C$ are not both 0 .
Show that if $A \neq C$, but $A C>0$, then the equation is equivalent to an equation of one of the forms

$$
\begin{aligned}
& \quad \frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1 \quad \text { or } \quad \frac{(y-k)^2}{a^2}+\frac{(x-h)^2}{b^2}=1 \\
& a>b>0, \text { provided } A\left(4 A C F-C D^2-A E^2\right)<0
\end{aligned}
$$

Nick Johnson

Numerade Educator

Problem 43

refer to the equation $A x^2+C y^2+D x+E y+F=0$, where $A$ and $C$ are not both 0 .
Show that if $A C<0$, then the equation is equivalent to an equation of one of the forms

$$
\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1 \quad \text { or } \quad \frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1 \text {, }
$$

$a>0, b>0$, provided $4 A C F-C D^2-A E^2 \neq 0$.

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Problem 44

refer to the equation $A x^2+C y^2+D x+E y+F=0$, where $A$ and $C$ are not both 0 .
In Exercise 42 , why must $A\left(4 A C F-C D^2-A E^2\right)$ be negative? What can be said about the graph if $A\left(4 A C F-C D^2-A E^2\right)$ is 0 ? What if it is positive?

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 45

refer to the equation $A x^2+C y^2+D x+E y+F=0$, where $A$ and $C$ are not both 0 .
In Exercise 43, why must $4 A C F-C D^2-A E^2$ not equal zero? What can be said about the graph if $4 A C F-C D^2-A E^2=0$ ?

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 46

By a translation of axes, points $\mathbf{S}\left(x_1, y_1\right)$ and $\mathbf{T}\left(x_2, y_2\right)$ are given new coordinates $\left(x_1^{\prime}, y_1^{\prime}\right)$ and $\left(x_2^{\prime}, y_2^{\prime}\right)$, respectively, in accordance with Equations (2) on page 171. Verify algebraically that

$$
\sqrt{\left(x_2^{\prime}-x_1^{\prime}\right)^2+\left(y_2^{\prime}-y_1^{\prime}\right)^2}=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}
$$

and interpret this fact geometrically.

Carson Merrill

Numerade Educator

Problem 47

By a translation of axes, (distinct) points $\mathbf{S}\left(x_1, y_1\right), \mathbf{T}\left(x_2, y_2\right)$, and $\mathbf{U}\left(x_3, y_3\right)$ are given new coordinates $\mathbf{S}^{\prime}\left(x_1^{\prime}, y_1^{\prime}\right), \mathbf{T}^{\prime}\left(x_2^{\prime}, y_2^{\prime}\right)$, and $\mathbf{U}^{\prime}\left(x_3^{\prime}, y_3^{\prime}\right)$, respectively. If $\mathbf{s}, \mathbf{t}, \mathbf{u}, \mathbf{s}^{\prime}, \mathbf{t}^{\prime}$, and $\mathbf{u}^{\prime}$ are the vectors corresponding to $\mathbf{S}, \mathbf{T}, \mathbf{U}, \mathbf{S}^{\prime}, \mathbf{T}^{\prime}$, and $\mathbf{U}^{\prime}$, respectively, verify algebraically that

$$
\frac{\left(\mathbf{t}^{\prime}-\mathbf{s}^{\prime}\right) \cdot\left(\mathbf{u}^{\prime}-\mathbf{s}^{\prime}\right)}{\left\|\mathbf{t}^{\prime}-\mathbf{s}^{\prime}\right\|\left\|\mathbf{u}^{\prime}-\mathbf{s}^{\prime}\right\|}=\frac{(\mathbf{t}-\mathbf{s}) \cdot(\mathbf{u}-\mathbf{s})}{\|\mathbf{t}-\mathbf{s}\|\|\mathbf{u}-\mathbf{s}\|}
$$

and interpret this fact geometrically.

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