Question 1

Select the curve represented by the parametric equations. ​
Prolate cycloid: $$x = \Theta - \frac { 4 } { 3 } \sin \Theta , y = 1 - \frac { 4 } { 3 } \cos \theta$$ ​

Accepted Answer

A) ​   
B) ​   
C) ​   
D) ​   
E) ​   
A) ​   
B) ​   
C) ​   
D) ​   
E) ​   A

Question 2

Find a set of parametric equations for the rectangular equation.​ $\begin{array} { l } 
t = 5 - x \
y = x ^ { 2 } - 3
\end{array}$ ​

Accepted Answer

A)  $x = t - 5 , y = t ^ { 2 } + 10 t - 22$ 
B)  $x = t - 5 , y = t ^ { 2 } - 10 t + 22$ 
C)  $x = - t + 5 , y = t ^ { 2 } - 22 t + 10$ 
D)  $x = - t + 5 , y = t ^ { 2 } - 10 t + 22$ 
E)  $x = t + 5 , y = t ^ { 2 } - 10 t + 22$ 
A)  $x = t - 5 , y = t ^ { 2 } + 10 t - 22$ 
B)  $x = t - 5 , y = t ^ { 2 } - 10 t + 22$ 
C)  $x = - t + 5 , y = t ^ { 2 } - 22 t + 10$ 
D)  $x = - t + 5 , y = t ^ { 2 } - 10 t + 22$ 
E)  $x = t + 5 , y = t ^ { 2 } - 10 t + 22$ D

Question 3

Select the parametric equations matching with the following graph.​   ​

Accepted Answer

A) Lissajous curve: $x = 2 \cos \theta , y = 2 \sin \theta$ 
B) Lissajous curve: $x = 2 \cos 2 \theta , y = 2 \sin 2 \theta$ 
C) Lissajous curve: $x = 2 \cos \theta , y = 2 \sin 2 \theta$ 
D) Lissajous curve: $x = 2 \cos \theta , y = \sin 2 \theta$ , 
E) Lissajous curve: $x = 2 \cos 2 \theta , y = \sin 2 \theta$ 
A) Lissajous curve: $x = 2 \cos \theta , y = 2 \sin \theta$ 
B) Lissajous curve: $x = 2 \cos 2 \theta , y = 2 \sin 2 \theta$ 
C) Lissajous curve: $x = 2 \cos \theta , y = 2 \sin 2 \theta$ 
D) Lissajous curve: $x = 2 \cos \theta , y = \sin 2 \theta$ , 
E) Lissajous curve: $x = 2 \cos 2 \theta , y = \sin 2 \theta$ D

Question 4

A projectile is launched at a height of h feet above the ground at an angle of $\theta$ with the horizontal.The initial velocity is $v _ { 0 }$ feet per second,and the path of the projectile is modeled by the parametric equations $x = \left( v _ { 0 } \cos \theta \right) t \text { and } y = h + \left( v _ { 0 } \sin \theta \right) t - 16 t ^ { 2 }$ . Select the correct graph of the path of a projectile launched from ground level at the value of $\theta$ and $v _ { 0 }$ .​ $\theta = 15 ^ { \circ } , v _ { 0 } = 80$ feet per second
​

Accepted Answer

A) ​   ​ 
B) ​   ​ 
C) ​   ​ 
D) ​   ​ 
E) ​   ​
​ 
A) ​   ​ 
B) ​   ​ 
C) ​   ​ 
D) ​   ​ 
E) ​   ​
​

Question 5

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.​ $x = 3 ( t + 1 )$ $y = | t - 3 |$ ​

Accepted Answer

A)  $y = \left| \frac { x } { 4 } - 3 \right|$ 
B)  $y = \left| \frac { x } { 3 } + 4 \right|$ 
C)  $y = | t - 4 |$ 
D)  $$y = \left| \frac { x } { 3 } - 4 \right|$$ 
E)  $$y = | t + 4 |$$ 
A)  $y = \left| \frac { x } { 4 } - 3 \right|$ 
B)  $y = \left| \frac { x } { 3 } + 4 \right|$ 
C)  $y = | t - 4 |$ 
D)  $$y = \left| \frac { x } { 3 } - 4 \right|$$ 
E)  $$y = | t + 4 |$$

Question 6

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.​ $x = \cos \theta$ $y = 5 \sin 2 \theta$ ​

Accepted Answer

A)  $\frac { x ^ { 2 } } { 1 } + \frac { y ^ { 2 } } { 5 } = 1$ 
B)  $y = \pm 10 x \sqrt { 1 + x ^ { 2 } }$ 
C)  $y = \pm 10 x \sqrt { 1 + x }$ 
D) ​ $\frac { x ^ { 2 } } { 1 } - \frac { y ^ { 2 } } { 5 } = 1$ 
E)  $y = \pm 10 x \sqrt { 1 - x ^ { 2 } }$ 
A)  $\frac { x ^ { 2 } } { 1 } + \frac { y ^ { 2 } } { 5 } = 1$ 
B)  $y = \pm 10 x \sqrt { 1 + x ^ { 2 } }$ 
C)  $y = \pm 10 x \sqrt { 1 + x }$ 
D) ​ $\frac { x ^ { 2 } } { 1 } - \frac { y ^ { 2 } } { 5 } = 1$ 
E)  $y = \pm 10 x \sqrt { 1 - x ^ { 2 } }$

Question 7

Select the curve represented by the parametric equations.​ $x = t + 1$ $y = \frac { t } { t + 1 }$ ​

Accepted Answer

A) ​   ​ 
B) ​   
C) ​   
D) ​   
E) ​   
A) ​   ​ 
B) ​   
C) ​   
D) ​   
E) ​

Question 8

Select the curve represented by the parametric equations.​ $x = t + 2$ $y = t ^ { 2 }$ ​

Accepted Answer

A) ​   
B) ​   
C) ​   
D) ​   
E) ​   
A) ​   
B) ​   
C) ​   
D) ​   
E) ​

Question 9

Select the curve represented by the parametric equations.(indicate the orientation of the curve)​ $$\begin{array} { l } 
x = 3 - 4 t \
y = 4 + 3 t
\end{array}$$ ​

Accepted Answer

A) ​   
B) ​   
C) ​   
D) ​   
E) ​   
A) ​   
B) ​   
C) ​   
D) ​   
E) ​

Question 10

Using following result find a set of parametric equation of the line.​ $x = x _ { 1 } + t \left( x _ { 2 } - x _ { 1 } \right) , y = y _ { 1 } + t \left( y _ { 2 } - y _ { 1 } \right)$ ​ Line: passes through $( 9,2 )$ and $( - 3,9 )$ ​

Accepted Answer

A)  $x = 9 + 12 t , y = 2 + 7 t$ 
B)  $x = 2 - 12 t , y = 2 + 7 t$ , 
C)  $x = 2 - 12 t , y = 9 - 7 t$ , 
D)  $x = 2 - 12 t , y = 9 + 7 t$ 
E)  $x = 9 - 12 t , y = 2 + 7 t$ 
A)  $x = 9 + 12 t , y = 2 + 7 t$ 
B)  $x = 2 - 12 t , y = 2 + 7 t$ , 
C)  $x = 2 - 12 t , y = 9 - 7 t$ , 
D)  $x = 2 - 12 t , y = 9 + 7 t$ 
E)  $x = 9 - 12 t , y = 2 + 7 t$

Question 11

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.​ $x = t + 2$ $y = t ^ { 2 }$ ​

Accepted Answer

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

Question 12

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.​ $x = 1 + 3 \cos \theta$ $$y = 1 + 5 \sin \theta$$ ​

Accepted Answer

A)  $\frac { ( x - 1 ) ^ { 2 } } { 9 } + \frac { ( y - 1 ) ^ { 2 } } { 25 } = 1$ 
B)  $$\frac { ( x - 1 ) ^ { 2 } } { 9 } - \frac { ( y - 1 ) ^ { 2 } } { 25 } = 1$$ 
C)  $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 25 } = 1$ 
D) ​ $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 25 } = 1$ 
E)  $\frac { ( x - 1 ) ^ { 2 } } { 25 } - \frac { ( y - 1 ) ^ { 2 } } { 9 } = 1$ 
A)  $\frac { ( x - 1 ) ^ { 2 } } { 9 } + \frac { ( y - 1 ) ^ { 2 } } { 25 } = 1$ 
B)  $$\frac { ( x - 1 ) ^ { 2 } } { 9 } - \frac { ( y - 1 ) ^ { 2 } } { 25 } = 1$$ 
C)  $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 25 } = 1$ 
D) ​ $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 25 } = 1$ 
E)  $\frac { ( x - 1 ) ^ { 2 } } { 25 } - \frac { ( y - 1 ) ^ { 2 } } { 9 } = 1$

Question 13

Using following result find a set of parametric equation of the line.​ $$x = x _ { 1 } + t \left( x _ { 2 } - x _ { 1 } \right) , y = y _ { 1 } + t \left( y _ { 2 } - y _ { 1 } \right)$$ ​ Line: passes through (0,0)and $$( 5,8 )$$ ​

Accepted Answer

A)  $x = 8 t , y = 5 t$ 
B)  $$x = 5 t , y = - 8 t$$ 
C)  $$x = - 5 t , y = - t$$ 
D)  $x = - 5 t , y = 8 t$ 
E)  $x = 5 t , y = 8 t$ 
A)  $x = 8 t , y = 5 t$ 
B)  $$x = 5 t , y = - 8 t$$ 
C)  $$x = - 5 t , y = - t$$ 
D)  $x = - 5 t , y = 8 t$ 
E)  $x = 5 t , y = 8 t$

Question 14

Select the parametric equations matching with the following graph.​   ​

Accepted Answer

A) Serpentine curve: $x = \frac { 1 } { 4 } \cot \Theta , y = 4 \sin \Theta \cos \Theta$ 
B) Serpentine curve: $$x = \frac { 1 } { 5 } \tan \Theta , y = 5 \sin \Theta \cos \theta$$ 
C) Serpentine curve: $x = \frac { 1 } { 5 } \cot \Theta , y = 4 \sin \Theta \cos \Theta$ 
D) Serpentine curve: $$x = \frac { 1 } { 4 } \cot \Theta , y = 5 \sin \Theta \cos \Theta$$ 
E) Serpentine curve: $x = \tan \theta , y = \sin \theta$ 
A) Serpentine curve: $x = \frac { 1 } { 4 } \cot \Theta , y = 4 \sin \Theta \cos \Theta$ 
B) Serpentine curve: $$x = \frac { 1 } { 5 } \tan \Theta , y = 5 \sin \Theta \cos \theta$$ 
C) Serpentine curve: $x = \frac { 1 } { 5 } \cot \Theta , y = 4 \sin \Theta \cos \Theta$ 
D) Serpentine curve: $$x = \frac { 1 } { 4 } \cot \Theta , y = 5 \sin \Theta \cos \Theta$$ 
E) Serpentine curve: $x = \tan \theta , y = \sin \theta$

Question 15

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.​ $x = \sqrt { t }$ $y = 4 - t$ ​

Accepted Answer

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

Question 16

Select the curve represented by the parametric equations.​ $x = t - 1$ $y = \frac { t } { t - 1 }$ ​

Accepted Answer

A) ​   
B) ​   
C) ​   
D) ​ ​   
E) ​   
A) ​   
B) ​   
C) ​   
D) ​ ​   
E) ​

Question 17

Select the curve represented by the parametric equations. ​
Cycloid: $x = \Theta + \sin \theta , y = 4 - \cos \theta$ ​

Accepted Answer

A) ​   
B) ​   
C) ​   
D) ​   
E) ​   
A) ​   
B) ​   
C) ​   
D) ​   
E) ​

Question 18

Find a set of parametric equations for the rectangular equation.​ $t = 2 - x$ $y = x ^ { 2 } + 5$ ​

Accepted Answer

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

Question 19

Using following result find a set of parametric equation of conic. ​
Hyperbola: $x = h + a \sec \theta , y = k + b \tan \theta$ ​
Hyperbola: vertices: $( \pm 4,0 )$ ;foci: $( \pm 5,0 )$ ​

Accepted Answer

A)  $x = 5 + 4 \sec \theta , y = 5 + 3 \tan \theta$ 
B)  $x = 4 \sec \theta , y = 3 \tan \theta$ 
C)  $x = 5 - 4 \sec \theta , y = 3 \tan \theta$ 
D)  $x = 3 \sec \theta , y = 4 \tan \theta$ 
E)  $$x = 5 \sec \theta , y = 4 \tan \theta$$ 
A)  $x = 5 + 4 \sec \theta , y = 5 + 3 \tan \theta$ 
B)  $x = 4 \sec \theta , y = 3 \tan \theta$ 
C)  $x = 5 - 4 \sec \theta , y = 3 \tan \theta$ 
D)  $x = 3 \sec \theta , y = 4 \tan \theta$ 
E)  $$x = 5 \sec \theta , y = 4 \tan \theta$$

Question 20

Select the curve represented by the parametric equations. ​
Prolate cycloid: $x = 3 \Theta - 7 \sin \theta , y = 3 - 7 \cos \theta$ ​

Accepted Answer

A) ​   
B) ​   
C) ​   ​ 
D) ​   ​ 
E) ​   
A) ​   
B) ​   
C) ​   ​ 
D) ​   ​ 
E) ​

Exam 64: Parametric Equations

Select the curve represented by the parametric equations. ​ Prolate cycloid: x=Θ−43sin⁡Θ,y=1−43cos⁡θx = \Theta - \frac { 4 } { 3 } \sin \Theta , y = 1 - \frac { 4 } { 3 } \cos \thetax=Θ−34​sinΘ,y=1−34​cosθ ​

Find a set of parametric equations for the rectangular equation.​ t=5-x y=-3 ​

Select the parametric equations matching with the following graph.​ ​

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.​ x=3(t+1)x = 3 ( t + 1 )x=3(t+1) y=∣t−3∣y = | t - 3 |y=∣t−3∣ ​

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.​ x=cos⁡θx = \cos \thetax=cosθ y=5sin⁡2θy = 5 \sin 2 \thetay=5sin2θ ​

Select the curve represented by the parametric equations.​ x=t+1x = t + 1x=t+1 y=tt+1y = \frac { t } { t + 1 }y=t+1t​ ​

Select the curve represented by the parametric equations.​ x=t+2x = t + 2x=t+2 y=t2y = t ^ { 2 }y=t2 ​

Select the curve represented by the parametric equations.(indicate the orientation of the curve)​ x=3-4t y=4+3t ​

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.​ x=t+2x = t + 2x=t+2 y=t2y = t ^ { 2 }y=t2 ​

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.​ x=1+3cos⁡θx = 1 + 3 \cos \thetax=1+3cosθ y=1+5sin⁡θy = 1 + 5 \sin \thetay=1+5sinθ ​

Select the parametric equations matching with the following graph.​ ​

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.​ x=tx = \sqrt { t }x=t​ y=4−ty = 4 - ty=4−t ​

Select the curve represented by the parametric equations.​ x=t−1x = t - 1x=t−1 y=tt−1y = \frac { t } { t - 1 }y=t−1t​ ​

Select the curve represented by the parametric equations. ​ Cycloid: x=Θ+sin⁡θ,y=4−cos⁡θx = \Theta + \sin \theta , y = 4 - \cos \thetax=Θ+sinθ,y=4−cosθ ​

Find a set of parametric equations for the rectangular equation.​ t=2−xt = 2 - xt=2−x y=x2+5y = x ^ { 2 } + 5y=x2+5 ​

Using following result find a set of parametric equation of conic. ​ Hyperbola: x=h+asec⁡θ,y=k+btan⁡θx = h + a \sec \theta , y = k + b \tan \thetax=h+asecθ,y=k+btanθ ​ Hyperbola: vertices: (±4,0)( \pm 4,0 )(±4,0) ;foci: (±5,0)( \pm 5,0 )(±5,0) ​

Select the curve represented by the parametric equations. ​ Prolate cycloid: x=3Θ−7sin⁡θ,y=3−7cos⁡θx = 3 \Theta - 7 \sin \theta , y = 3 - 7 \cos \thetax=3Θ−7sinθ,y=3−7cosθ ​

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Select the curve represented by the parametric equations. Prolate cycloid: $x = \Theta - \frac { 4 } { 3 } \sin \Theta , y = 1 - \frac { 4 } { 3 } \cos \theta$

Find a set of parametric equations for the rectangular equation. t=5-x y=-3

Select the parametric equations matching with the following graph.

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. $x = 3 ( t + 1 )$ $y = | t - 3 |$

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. $x = \cos \theta$ $y = 5 \sin 2 \theta$

Select the curve represented by the parametric equations. $x = t + 1$ $y = \frac { t } { t + 1 }$

Select the curve represented by the parametric equations. $x = t + 2$ $y = t ^ { 2 }$

Select the curve represented by the parametric equations.(indicate the orientation of the curve) x=3-4t y=4+3t

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. $x = t + 2$ $y = t ^ { 2 }$

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. $x = 1 + 3 \cos \theta$ $y = 1 + 5 \sin \theta$

Select the parametric equations matching with the following graph.

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. $x = \sqrt { t }$ $y = 4 - t$

Select the curve represented by the parametric equations. $x = t - 1$ $y = \frac { t } { t - 1 }$

Select the curve represented by the parametric equations. Cycloid: $x = \Theta + \sin \theta , y = 4 - \cos \theta$

Find a set of parametric equations for the rectangular equation. $t = 2 - x$ $y = x ^ { 2 } + 5$

Using following result find a set of parametric equation of conic. Hyperbola: $x = h + a \sec \theta , y = k + b \tan \theta$ Hyperbola: vertices: $( \pm 4,0 )$ ;foci: $( \pm 5,0 )$

Select the curve represented by the parametric equations. Prolate cycloid: $x = 3 \Theta - 7 \sin \theta , y = 3 - 7 \cos \theta$