Question 1

Find a polar equation in r and $\theta$ that has the same graph as the equation in x and y. $x = - 6$

Accepted Answer

A)  $r = - 6 \sin \theta$ 
B)  $r = - 6 \sec \theta$ 
C)  $r = - 6$ 
D)  $r = - 6 \cos \theta$ 
E)  $r = 6 \csc \theta$ 
A)  $r = - 6 \sin \theta$ 
B)  $r = - 6 \sec \theta$ 
C)  $r = - 6$ 
D)  $r = - 6 \cos \theta$ 
E)  $r = 6 \csc \theta$ B

Question 2

Find a polar equation of the conic with focus at the pole that has the eccentricity and equation of directrix. $e = 4 , \quad r = - 2 \csc \theta$

Accepted Answer

A)  $r = \frac { 8 } { 1 - 4 \sin \theta }$ 
B)  $r = \frac { 8 } { 2 + \sin \theta }$ 
C)  $r = \frac { 8 } { 1 - 4 \cos \theta }$ 
D)  $r = \frac { 2 } { 4 + \cos \theta }$ 
E)  $r = \frac { 8 } { 1 + 4 \sin \theta }$ 
A)  $r = \frac { 8 } { 1 - 4 \sin \theta }$ 
B)  $r = \frac { 8 } { 2 + \sin \theta }$ 
C)  $r = \frac { 8 } { 1 - 4 \cos \theta }$ 
D)  $r = \frac { 2 } { 4 + \cos \theta }$ 
E)  $r = \frac { 8 } { 1 + 4 \sin \theta }$ A

Question 3

An arch of a bridge is semi-elliptical, with major axis horizontal. The base of the arch is 30 feet across, and the highest part of the arch is 10 feet above the horizontal roadway, as shown in the figure. Find the height of the arch 7 feet from the center of the base.

Accepted Answer

A)  12.1 ft 
B)  8.8 ft 
C)  17.7 ft 
D)  11.0 ft 
E)  7.3 ft 
A)  12.1 ft 
B)  8.8 ft 
C)  17.7 ft 
D)  11.0 ft 
E)  7.3 ft B

Question 4

Find an equation for the ellipse that has its center at the origin and satisfies the conditions. $\text { vertices } V ( \pm 7,0 ) \text {, foci } F ( \pm 3,0 )$

Accepted Answer

A)  $\frac { x ^ { 2 } } { 58 } + \frac { y ^ { 2 } } { 40 } = 1$ 
B)  $\frac { x ^ { 2 } } { 40 } + \frac { y ^ { 2 } } { 49 } = 1$ 
C)  $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 40 } = 1$ 
D)  $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 58 } = 1$ 
E)  $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 40 } = 1$ 
A)  $\frac { x ^ { 2 } } { 58 } + \frac { y ^ { 2 } } { 40 } = 1$ 
B)  $\frac { x ^ { 2 } } { 40 } + \frac { y ^ { 2 } } { 49 } = 1$ 
C)  $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 40 } = 1$ 
D)  $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 58 } = 1$ 
E)  $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 40 } = 1$

Question 5

Find a polar equation in r and $\theta$ that has the same graph as the equation in x and y. $( x - 8 ) ^ { 2 } + y ^ { 2 } = 64$

Accepted Answer

A)  $r = 16 \cos 16 \theta$ 
B)  $r = \cos 16 \theta$ 
C)  $r = 16 \cos \theta$ 
D)  $r = 64 \sec \theta$ 
E)  $r = 16 \sin \theta$ 
A)  $r = 16 \cos 16 \theta$ 
B)  $r = \cos 16 \theta$ 
C)  $r = 16 \cos \theta$ 
D)  $r = 64 \sec \theta$ 
E)  $r = 16 \sin \theta$

Question 6

Find the foci of the ellipse. $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 9 } = 1$

Accepted Answer

A)  $( 0 , \sqrt { 40 } )$ 
B)  $( \sqrt { 40 } , 0 ) , ( - \sqrt { 40 } , 0 )$ 
C)  $( \sqrt { 58 } , 0 )$ 
D)  $( 0 , \sqrt { 40 } ) , ( - \sqrt { 40 } , 0 )$ 
E)  $( 0 , - \sqrt { 58 } ) , ( 0 , \sqrt { 58 } )$ 
A)  $( 0 , \sqrt { 40 } )$ 
B)  $( \sqrt { 40 } , 0 ) , ( - \sqrt { 40 } , 0 )$ 
C)  $( \sqrt { 58 } , 0 )$ 
D)  $( 0 , \sqrt { 40 } ) , ( - \sqrt { 40 } , 0 )$ 
E)  $( 0 , - \sqrt { 58 } ) , ( 0 , \sqrt { 58 } )$

Question 7

Find an equation in x and y for the polar equation. $r = \frac { 14 } { 12 + 4 \sin \theta }$

Accepted Answer

A)  $127 x ^ { 2 } + 146 y ^ { 2 } - 110 x + 194 = 0$ 
B)  $144 x ^ { 2 } + 128 y ^ { 2 } + 112 y - 196 = 0$ 
C)  $146 x ^ { 2 } - 127 y ^ { 2 } + 110 y + 194 = 0$ 
D)  $128 x ^ { 2 } + 144 y ^ { 2 } + 112 x - 196 = 0$ 
E)  $128 x ^ { 2 } + 144 y ^ { 2 } + 112 y - 196 = 0$ 
A)  $127 x ^ { 2 } + 146 y ^ { 2 } - 110 x + 194 = 0$ 
B)  $144 x ^ { 2 } + 128 y ^ { 2 } + 112 y - 196 = 0$ 
C)  $146 x ^ { 2 } - 127 y ^ { 2 } + 110 y + 194 = 0$ 
D)  $128 x ^ { 2 } + 144 y ^ { 2 } + 112 x - 196 = 0$ 
E)  $128 x ^ { 2 } + 144 y ^ { 2 } + 112 y - 196 = 0$

Question 8

Find a polar equation in r and $\theta$ that has the same graph as the equation in x and y. $3 y = - x$

Accepted Answer

A)  $\theta = \cot ^ { - 1 } \left( - \frac { 1 } { 3 } \right)$ 
B)  $r = \frac { 3 } { \sin \theta }$ 
C)  $r = \frac { 3 } { \cos \theta }$ 
D)  $\theta = \tan 3$ 
E)  $\theta = \tan ^ { - 1 } \left( - \frac { 1 } { 3 } \right)$ 
A)  $\theta = \cot ^ { - 1 } \left( - \frac { 1 } { 3 } \right)$ 
B)  $r = \frac { 3 } { \sin \theta }$ 
C)  $r = \frac { 3 } { \cos \theta }$ 
D)  $\theta = \tan 3$ 
E)  $\theta = \tan ^ { - 1 } \left( - \frac { 1 } { 3 } \right)$

Question 9

Polar equations of conics can be used to describe the motion of comets. These paths can be graphed using the following polar equation, where e is the eccentricity of the conic and r per  is the perihelion distance measured in AU. For Encke's Comet r per  = 0.3317 and e = 0.8499, determine whether its trajectory is elliptical, parabolic, or hyperbolic. $r = \frac { r _ { \mathrm { per } } ( 1 + e ) } { 1 - e \cos \theta }$

Accepted Answer

A)  parabolic 
B)  elliptical 
C)  hyperbolic 
A)  parabolic 
B)  elliptical 
C)  hyperbolic

Question 10

Find an equation of the parabola that satisfies the condition. Vertex $V ( - 4,0 )$ , focus $F ( - 2,0 )$

Accepted Answer

A)  $y = 8 ( x + 4 ) ^ { 2 }$ 
B)  $y ^ { 2 } = 8 ( x - 4 )$ 
C)  $y ^ { 2 } = 8 ( x + 4 )$ 
D)  $y ^ { 2 } = ( x + 4 )$ 
E)  $x ^ { 2 } = 8 ( y + 4 )$ 
A)  $y = 8 ( x + 4 ) ^ { 2 }$ 
B)  $y ^ { 2 } = 8 ( x - 4 )$ 
C)  $y ^ { 2 } = 8 ( x + 4 )$ 
D)  $y ^ { 2 } = ( x + 4 )$ 
E)  $x ^ { 2 } = 8 ( y + 4 )$

Question 11

Find an equation for the parabola that has a horizontal axis and passes through the given points. $P ( 34 , - 2 ) , Q ( 34,5 ) , R ( - 2,2 )$

Accepted Answer

A)  $x = 9 y ^ { 2 } + 3 y + 4$ 
B)  $x = 3 y ^ { 2 } + 4 y + 9$ 
C)  $x = - 3 y ^ { 2 } - 9 y + 4$ 
D)  $x = 3 y ^ { 2 } + 9 y - 4$ 
E)  $x = 3 y ^ { 2 } - 9 y + 4$ 
A)  $x = 9 y ^ { 2 } + 3 y + 4$ 
B)  $x = 3 y ^ { 2 } + 4 y + 9$ 
C)  $x = - 3 y ^ { 2 } - 9 y + 4$ 
D)  $x = 3 y ^ { 2 } + 9 y - 4$ 
E)  $x = 3 y ^ { 2 } - 9 y + 4$

Question 12

Find the foci of the ellipse. $\frac { ( x + 1 ) ^ { 2 } } { 25 } + \frac { ( y - 2 ) ^ { 2 } } { 4 } = 1$

Accepted Answer

A)  $$( \sqrt { 21 } + 1,2 )$$ 
B)  $( \sqrt { 21 } \pm 1,2 )$ 
C)  $( \sqrt { 21 } - 1 , - 2 )$ 
D)  $( \sqrt { 21 } + 1 , \pm 2 )$ 
E)  $( \pm \sqrt { 21 } - 1,2 )$ 
A)  $$( \sqrt { 21 } + 1,2 )$$ 
B)  $( \sqrt { 21 } \pm 1,2 )$ 
C)  $( \sqrt { 21 } - 1 , - 2 )$ 
D)  $( \sqrt { 21 } + 1 , \pm 2 )$ 
E)  $( \pm \sqrt { 21 } - 1,2 )$

Question 13

Find the eccentricity. $r = \csc \theta ( \csc \theta + \cot \theta )$

Accepted Answer

A)  $$e = 4$$ 
B)  $e = \frac { 1 } { 7 }$ 
C)  $$e = 1$$ 
D)  $e = 3$ 
E)  $e = \frac { 1 } { 2 }$ 
A)  $$e = 4$$ 
B)  $e = \frac { 1 } { 7 }$ 
C)  $$e = 1$$ 
D)  $e = 3$ 
E)  $e = \frac { 1 } { 2 }$

Question 14

Classify the conic. $r = \frac { 6 } { 2 - 2 \sin \theta }$

Accepted Answer

A)  parabola 
B)  ellipse 
C)  circle 
D)  hyperbola 
A)  parabola 
B)  ellipse 
C)  circle 
D)  hyperbola

Question 15

Sketch the graph of the ellipse, showing the foci . $4 x ^ { 2 } + y ^ { 2 } = 6 y - 8$

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 16

Find a polar equation of the conic with focus at the pole that has the eccentricity and equation of directrix. $e = 1$, $r \cos \theta = 7$

Accepted Answer

A)  $r = \frac { 7 } { 1 + 2 \cos \theta }$ 
B)  $r = \frac { 3 } { 1 + \sin \theta }$ 
C)  $r = \frac { 7 } { 1 - \cos \theta }$ 
D)  $r = \frac { 7 } { 1 + \cos \theta }$ 
E)  $r = \frac { 3 } { 1 - \cos \theta }$ 
A)  $r = \frac { 7 } { 1 + 2 \cos \theta }$ 
B)  $r = \frac { 3 } { 1 + \sin \theta }$ 
C)  $r = \frac { 7 } { 1 - \cos \theta }$ 
D)  $r = \frac { 7 } { 1 + \cos \theta }$ 
E)  $r = \frac { 3 } { 1 - \cos \theta }$

Question 17

Find a polar equation in r and 0 of the parabola with focus at the pole and the given vertex. $V \left( 9 , \frac { \pi } { 2 } \right)$

Accepted Answer

A)  $r = \frac { 18 } { 1 - \cos \theta }$ 
B)  $r = \frac { 18 } { 1 + \sin \theta }$ 
C)  $r = \frac { 9 } { 1 + 9 \sin \theta }$ 
D)  $r = \frac { 9 } { 1 + \cos \theta }$ 
E)  $r = \frac { 16 } { 1 - \sin \theta }$ 
A)  $r = \frac { 18 } { 1 - \cos \theta }$ 
B)  $r = \frac { 18 } { 1 + \sin \theta }$ 
C)  $r = \frac { 9 } { 1 + 9 \sin \theta }$ 
D)  $r = \frac { 9 } { 1 + \cos \theta }$ 
E)  $r = \frac { 16 } { 1 - \sin \theta }$

Question 18

Find a polar equation in r and 0 of the parabola with focus at the pole and the given vertex. $V ( 5,0 )$

Accepted Answer

A)  $r = \frac { 12 } { 1 - \sin \theta }$ 
B)  $r = \frac { 10 } { 1 + \cos \theta }$ 
C)  $r = \frac { 5 } { 1 + \sin \theta }$ 
D)  $r = \frac { 10 } { 1 + 5 \cos \theta }$ 
E)  $r = \frac { 10 } { 1 - \cos \theta }$ 
A)  $r = \frac { 12 } { 1 - \sin \theta }$ 
B)  $r = \frac { 10 } { 1 + \cos \theta }$ 
C)  $r = \frac { 5 } { 1 + \sin \theta }$ 
D)  $r = \frac { 10 } { 1 + 5 \cos \theta }$ 
E)  $r = \frac { 10 } { 1 - \cos \theta }$

Question 19

Shown in the figure is the Lissajous figure given by $x = 4 \sin 3 t , y = 4 \sin 1.5 t ; t \geq 0$ Find the period of the figure - that is, the length of the smallest t-interval that traces the curve.

Accepted Answer

A)  $\frac { 4 \pi } { 3 }$ 
B)  $$\frac { \pi } { 3 }$$ 
C)  π 
D)  $2 \pi$ 
E)  $\frac { 3 \pi } { 4 }$ 
A)  $\frac { 4 \pi } { 3 }$ 
B)  $$\frac { \pi } { 3 }$$ 
C)  π 
D)  $2 \pi$ 
E)  $\frac { 3 \pi } { 4 }$

Question 20

Lissajous figures are used in the study of electrical circuits to determine the phase difference $\varphi$ between a known voltage $V _ { 1 } ( t ) = A \sin ( a t )$ and an unknown voltage $V _ { 2 } ( t ) = B \sin ( ω t + \varphi )$
$a \mathrm{dt}$ having the same frequency. The voltages are graphed parametrically as $x = V _ { 1 } ( t )$ and $y = V _ { 2 } ( t )$ . If $\varphi$ is acute, then $\varphi = \sin ^ { - 1 } \frac { y _ { \mathrm { int } } } { y _ { \max } }$ where y int  is the nonnegative y -intercept and y max  is the maximum y -value on the curve. Graph the parametric curve and use the graph to approximate $\varphi$ in degrees if $\begin{array} { c } 
V _ { 1 } ( t ) = 40 \sin ( 60 \pi t ) , V _ { 2 } ( t ) = 90 \cos ( 60 \pi t - \pi / 3 ) , \
0 \leq t \leq 0.035
\end{array}$

Accepted Answer

A)  $30 ^ { \circ }$ 
B)  $90 ^ { \circ }$ 
C)  $60 ^ { \circ }$ 
D)  $45 ^ { \circ }$ 
E)  $15 ^ { \circ }$ 
A)  $30 ^ { \circ }$ 
B)  $90 ^ { \circ }$ 
C)  $60 ^ { \circ }$ 
D)  $45 ^ { \circ }$ 
E)  $15 ^ { \circ }$

Find a polar equation in r and $\theta$ that has the same graph as the equation in x and y. $x = - 6$

Find a polar equation of the conic with focus at the pole that has the eccentricity and equation of directrix. $e = 4 , \quad r = - 2 \csc \theta$

An arch of a bridge is semi-elliptical, with major axis horizontal. The base of the arch is 30 feet across, and the highest part of the arch is 10 feet above the horizontal roadway, as shown in the figure. Find the height of the arch 7 feet from the center of the base.

Find an equation for the ellipse that has its center at the origin and satisfies the conditions. $\text { vertices } V ( \pm 7,0 ) \text {, foci } F ( \pm 3,0 )$

Find a polar equation in r and $\theta$ that has the same graph as the equation in x and y. $( x - 8 ) ^ { 2 } + y ^ { 2 } = 64$

Find the foci of the ellipse. $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 9 } = 1$

Find an equation in x and y for the polar equation. $r = \frac { 14 } { 12 + 4 \sin \theta }$

Find a polar equation in r and $\theta$ that has the same graph as the equation in x and y. $3 y = - x$

Find an equation of the parabola that satisfies the condition. Vertex $V ( - 4,0 )$ , focus $F ( - 2,0 )$

Find an equation for the parabola that has a horizontal axis and passes through the given points. $P ( 34 , - 2 ) , Q ( 34,5 ) , R ( - 2,2 )$

Find the foci of the ellipse. $\frac { ( x + 1 ) ^ { 2 } } { 25 } + \frac { ( y - 2 ) ^ { 2 } } { 4 } = 1$

Find the eccentricity. $r = \csc \theta ( \csc \theta + \cot \theta )$

Classify the conic. $r = \frac { 6 } { 2 - 2 \sin \theta }$

Sketch the graph of the ellipse, showing the foci . $4 x ^ { 2 } + y ^ { 2 } = 6 y - 8$

Find a polar equation of the conic with focus at the pole that has the eccentricity and equation of directrix. $e = 1$ , $r \cos \theta = 7$

Find a polar equation in r and 0 of the parabola with focus at the pole and the given vertex. $V \left( 9 , \frac { \pi } { 2 } \right)$

Find a polar equation in r and 0 of the parabola with focus at the pole and the given vertex. $V ( 5,0 )$

Topics From Algebra

Functions and Graphs

Polynomial and Rational Functions

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometry

Systems of Equations and Inequalities

Sequences, Series, and Probability

Filters

Exam 10: Topics From Analytical Geometry

Find a polar equation in r and θ\thetaθ that has the same graph as the equation in x and y. x=−6x = - 6x=−6

Find a polar equation of the conic with focus at the pole that has the eccentricity and equation of directrix. e=4,r=−2csc⁡θe = 4 , \quad r = - 2 \csc \thetae=4,r=−2cscθ

An arch of a bridge is semi-elliptical, with major axis horizontal. The base of the arch is 30 feet across, and the highest part of the arch is 10 feet above the horizontal roadway, as shown in the figure. Find the height of the arch 7 feet from the center of the base.

Find an equation for the ellipse that has its center at the origin and satisfies the conditions. vertices V(±7,0), foci F(±3,0)\text { vertices } V ( \pm 7,0 ) \text {, foci } F ( \pm 3,0 ) vertices V(±7,0), foci F(±3,0)

Find a polar equation in r and θ\thetaθ that has the same graph as the equation in x and y. (x−8)2+y2=64( x - 8 ) ^ { 2 } + y ^ { 2 } = 64(x−8)2+y2=64

Find the foci of the ellipse. x249+y29=1\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 9 } = 149x2​+9y2​=1

Find an equation in x and y for the polar equation. r=1412+4sin⁡θr = \frac { 14 } { 12 + 4 \sin \theta }r=12+4sinθ14​

Find a polar equation in r and θ\thetaθ that has the same graph as the equation in x and y. 3y=−x3 y = - x3y=−x

Find an equation of the parabola that satisfies the condition. Vertex V(−4,0)V ( - 4,0 )V(−4,0) , focus F(−2,0)F ( - 2,0 )F(−2,0)

Find an equation for the parabola that has a horizontal axis and passes through the given points. P(34,−2),Q(34,5),R(−2,2)P ( 34 , - 2 ) , Q ( 34,5 ) , R ( - 2,2 )P(34,−2),Q(34,5),R(−2,2)

Find the foci of the ellipse. (x+1)225+(y−2)24=1\frac { ( x + 1 ) ^ { 2 } } { 25 } + \frac { ( y - 2 ) ^ { 2 } } { 4 } = 125(x+1)2​+4(y−2)2​=1

Find the eccentricity. r=csc⁡θ(csc⁡θ+cot⁡θ)r = \csc \theta ( \csc \theta + \cot \theta )r=cscθ(cscθ+cotθ)

Classify the conic. r=62−2sin⁡θr = \frac { 6 } { 2 - 2 \sin \theta }r=2−2sinθ6​

Sketch the graph of the ellipse, showing the foci . 4x2+y2=6y−84 x ^ { 2 } + y ^ { 2 } = 6 y - 84x2+y2=6y−8

Find a polar equation of the conic with focus at the pole that has the eccentricity and equation of directrix. e=1e = 1e=1 , rcos⁡θ=7r \cos \theta = 7rcosθ=7

Find a polar equation in r and 0 of the parabola with focus at the pole and the given vertex. V(9,π2)V \left( 9 , \frac { \pi } { 2 } \right)V(9,2π​)

Find a polar equation in r and 0 of the parabola with focus at the pole and the given vertex. V(5,0)V ( 5,0 )V(5,0)

Shown in the figure is the Lissajous figure given by x=4sin⁡3t,y=4sin⁡1.5t;t≥0x = 4 \sin 3 t , y = 4 \sin 1.5 t ; t \geq 0x=4sin3t,y=4sin1.5t;t≥0 Find the period of the figure - that is, the length of the smallest t-interval that traces the curve.

Topics From Algebra

Functions and Graphs

Polynomial and Rational Functions

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometry

Systems of Equations and Inequalities

Sequences, Series, and Probability

Filters

Find a polar equation in r and $\theta$ that has the same graph as the equation in x and y. $x = - 6$

Find a polar equation of the conic with focus at the pole that has the eccentricity and equation of directrix. $e = 4 , \quad r = - 2 \csc \theta$

Find an equation for the ellipse that has its center at the origin and satisfies the conditions. $\text { vertices } V ( \pm 7,0 ) \text {, foci } F ( \pm 3,0 )$

Find a polar equation in r and $\theta$ that has the same graph as the equation in x and y. $( x - 8 ) ^ { 2 } + y ^ { 2 } = 64$

Find the foci of the ellipse. $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 9 } = 1$

Find an equation in x and y for the polar equation. $r = \frac { 14 } { 12 + 4 \sin \theta }$

Find a polar equation in r and $\theta$ that has the same graph as the equation in x and y. $3 y = - x$

Find an equation of the parabola that satisfies the condition. Vertex $V ( - 4,0 )$ , focus $F ( - 2,0 )$

Find an equation for the parabola that has a horizontal axis and passes through the given points. $P ( 34 , - 2 ) , Q ( 34,5 ) , R ( - 2,2 )$

Find the foci of the ellipse. $\frac { ( x + 1 ) ^ { 2 } } { 25 } + \frac { ( y - 2 ) ^ { 2 } } { 4 } = 1$

Find the eccentricity. $r = \csc \theta ( \csc \theta + \cot \theta )$

Classify the conic. $r = \frac { 6 } { 2 - 2 \sin \theta }$

Sketch the graph of the ellipse, showing the foci . $4 x ^ { 2 } + y ^ { 2 } = 6 y - 8$

Find a polar equation of the conic with focus at the pole that has the eccentricity and equation of directrix. $e = 1$ , $r \cos \theta = 7$

Find a polar equation in r and 0 of the parabola with focus at the pole and the given vertex. $V \left( 9 , \frac { \pi } { 2 } \right)$

Find a polar equation in r and 0 of the parabola with focus at the pole and the given vertex. $V ( 5,0 )$