Deck 64: Parametric Equations

Question 1

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

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

Answer 1

$y = x ^ { 2 } - 4 x + 4$

Question 2

Select the curve represented by the parametric equations.​ $$\begin{array} { l } x = \frac { 2 } { 3 } t \\\\y = t ^ { 2 }\end{array}$$ ​ ​

A)​
B)​
C)​
D)​
E)​

Answer 2

​

Question 3

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

A)​
B)​
C)​
D)​
E)​

Answer 3

​

Question 4

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

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 5

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

A)​
B)​
C)​
D)​ ​
E)​

Question 6

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.​ $$x = t - 1$$ $$y = 5 t + 1$$ ​

A) $y = 5 x - 6$
B) $y = 5 x + 6$
C) $y = x + 5$
D) $y = x + 6$
E) $y = 5 x + 1$

Question 7

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.​ $\begin{array} { l } x = 5 \sin 2 \Theta \\y = 5 \cos 2 \theta\end{array}$ ​

A) $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 25 } = 1$
B) $\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 25 } = 1$
C) $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 2 } = 1$ .
D) $y = \frac { x } { 2 }$

Question 8

Select the curve represented by the parametric equations.​ $$\begin{array} { l } x = 5 \cos \theta \\y = 6 \sin \theta\end{array}$$ ​

A)​
B)​
C)​
D)​
E)​

Question 9

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

A) $y = x + 2$
B) $y = \frac { x + 2 } { x }$
C) $y = x - 2$
D) $y = \frac { x + 2 } { 3 x }$
E) $y = \frac { x } { x + 2 }$

Question 10

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.​ $$\begin{array} { l } x = \frac { 1 } { 4 } t \\\\y = t ^ { 2 }\end{array}$$ ​

A) $y = x ^ { 2 }$
B) $y = - 4 x ^ { 2 }$
C) $y = 4 x ^ { 2 }$
D) $y = 16 x ^ { 2 }$
E) $y = - 16 x ^ { 2 }$

Question 11

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

A)​ ​
B)​
C)​
D)​
E)​

Question 12

Select the curve represented by the parametric equations.​ $x = 1 + \cos \theta$ $y = 1 + 2 \sin \theta$ ​

A)​
B)​
C)​ ​
D)​
E)​

Question 13

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

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

Question 14

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.​ $x = t + 8$ $y = \frac { t } { t + 8 }$ ​

A) $y = \frac { x - 8 } { 9 x }$
B) $y = \frac { x } { x - 8 }$
C) $y = x - 8$
D) $y = \frac { x - 8 } { x }$
E) $y = x + 8$

Question 15

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.​ $\begin{array} { l } x = 4 \cos \theta \\y = 2 \sin \theta\end{array}$ ​

A)​ $$\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 16 } = 1$$
B) $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 4 } = 1$
C) $$y = \frac { x } { 4 }$$
D) $y = \frac { x } { 2 }$
E)​ $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 4 } = 1$

Question 16

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

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 17

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

A)​ ​
B)​ ​
C)​
D)​
E)​

Question 18

Select the curve represented by the parametric equations.(indicate the orientation of the curve)​ $x = t - 1$ $y = 4 t + 1$ ​

A)​
B)​
C)​
D)​
E)​

Question 19

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

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 20

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

A)​
B)​
C)​
D)​
E)​

Question 21

Using following result find a set of parametric equation of conic. ​
Circle: $x = h + r \cos \theta , y = k + r \sin \theta$ ​
Circle: center: $( 2,6 )$ ;radius: 8
​

A) $$x = 6 + 8 \cos \theta , y = 6 + 2 \sin \theta$$
B) $$x = 2 + 6 \cos \theta , y = 6 + 8 \sin \theta$$
C) $x = 2 + 8 \cos \theta , y = 6 + 8 \sin \theta$
D) $x = 8 + 2 \cos \theta , y = 6 + 2 \sin \theta$
E) $x = 8 + 6 \cos \theta , y = 6 + 2 \sin \theta$

Question 22

Using following result find a set of parametric equation of conic.​ $x = h + a \cos \theta , y = k + b i s n \theta$ ​ vertices: $( \pm 17,0 )$ ;foci: $( \pm 15,0 )$ ​

A) $x = 8 \cos \theta , y = 17 \sin \theta$
B) $x = 15 + 17 \cos \theta , y = 15 - 8 \sin \theta$
C) $x = 8 - 15 \cos \theta , y = 17 + 15 \sin \theta$
D) $x = 15 - 17 \cos \theta , y = 8 \sin \theta$
E) $x = 17 \cos \theta , y = 8 \sin \theta$

Question 23

Select the curve represented by the parametric equations. ​
Curtate cycloid: $$x = 4 \Theta - 2 \sin \theta , y = 4 - 2 \cos \theta$$ ​

A)​
B)​
C)​
D)​
E)​

Question 24

Find a set of parametric equations for the rectangular equation.​ $t = 2 - x$ $y = \frac { 1 } { 3 x }$ ​

A) $x = - 2 - t , y = \frac { 1 } { ( 6 - 6 t ) }$
B) $x = - 2 - t , y = \frac { 1 } { ( 6 - 3 t ) }$
C) $x = 2 - t , y = \frac { 1 } { ( 3 - 3 t ) }$
D) $x = - 2 + t , y = \frac { 1 } { ( 6 - 3 t ) }$
E) $x = 2 - t , y = \frac { 1 } { ( 6 - 3 t ) }$

Question 25

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

A)​
B)​
C)​ ​
D)​ ​
E)​

Question 26

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

A) $x = 5 + t , y = \frac { 1 } { 2 } ( 4 - t )$
B) $x = 2 - t , y = \frac { 1 } { 5 } ( 4 - t )$
C) $x = 2 + t , y = \frac { 1 } { 5 } ( 4 - t )$
D) $x = 2 + t , y = \frac { 1 } { 5 } ( 4 + t )$
E) $x = 2 + t , y = \frac { 1 } { 5 } ( t - 4 )$

Question 27

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

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) $x = t + 5 , y = t ^ { 2 } - 10 t + 22$

Question 28

Find a set of parametric equations for the rectangular equation.​ $t = 4 - x$ $y = \frac { 1 } { x }$ ​

A) $x = 4 - t , y = \frac { - 1 } { ( t + 4 ) }$
B) $x = 4 - t , y = \frac { - 1 } { ( - t - 4 ) }$
C) $x = 4 - t , y = \frac { - 1 } { ( t - 4 ) }$
D) $x = 4 + t , y = \frac { - 1 } { ( t - 8 ) }$
E) $x = 8 - t , y = \frac { - 1 } { ( t - 4 ) }$

Question 29

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

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 30

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 $$( 5,8 )$$ ​

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 31

Using following result find a set of parametric equation of conic. ​
Hyperbola: $x = h + a \sec \theta , y = k + b \tan \theta$ ​
Hyperbola: vertices: $( \pm 15,0 )$ ;foci: $( \pm 17,0 )$ ​

A) $x = 8 \sec \theta , y = 15 \tan \theta$
B) $x = 17 - 15 \sec \theta , y = 17 + 8 \tan \theta$
C) $x = 15 \sec \theta , y = 8 \tan \theta$
D) $x = - 8 + 15 \sec \theta , y = 17 - 8 \tan \theta$
E) $x = 17 \sec \theta , y = 17 \tan \theta$

Question 32

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

A)​
B)​
C)​
D)​
E)​

Question 33

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

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: $x = 2 \cos 2 \theta , y = \sin 2 \theta$

Question 34

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 $( - 3,9 )$ ​

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 35

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

A)​
B)​
C)​
D)​
E)​

Question 36

Find a set of parametric equations for the rectangular equation.​ $t = 2 - x$ $y = 4 x - 2$ ​

A) $x = - t + 4 y = - 4 t + 2$
B) $x = - t + 6 , y = - 4 t + 6$
C) $x = - t + 2 , y = - 6 t + 4$
D) $x = t + 2 , y = - 4 t + 6$
E) $x = - t + 2 , y = - 4 t + 6$

Question 37

Find a set of parametric equations for the rectangular equation.​ $\begin{array} { l } t = 2 - x \\y = 1 - 6 x ^ { 2 }\end{array}$ ​​

A) $x = - t + 2 , y = - 6 t ^ { 2 } + 24 t - 23$
B) $x = t + 2 , y = - 6 t ^ { 2 } + 24 t - 23$
C) $x = t - 2 , y = - 6 t ^ { 2 } + 24 t - 23$
D) $x = - t + 2 , y = - 6 t ^ { 2 } + 23 t - 24$
E) $x = t + 2 , y = - 24 t ^ { 2 } + 6 t + 23$

Question 38

Using following result find a set of parametric equation of conic. ​
Circle: $x = h + r \cos \theta , y = k + r \sin \theta$ ​
Circle: center: $( 8 , - 3 )$ ;radius: 5
​

A) $x = 5 + 3 \cos \theta , y = 8 - 3 \sin \theta$
B) $x = 3 + 5 \cos \theta , y = 8 - 5 \sin \theta$
C) $x = 8 + 5 \cos \theta , y = 3 - 5 \sin \theta$
D) $x = 8 - 5 \cos \theta , y = 3 + 5 \sin \theta$
E) $x = 8 + 5 \cos \theta , y = - 3 + 5 \sin \theta$

Question 39

Select the curve represented by the parametric equations. ​
Cycloid: $$x = 5 ( \Theta - \sin \theta ) , y = 5 ( 1 - \cos \theta )$$ ​

A)​
B)​
C)​
D)​
E)​

Question 40

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: $( \pm 5,0 )$ ​

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 41

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 $v _ { 0 }$ .​ $\theta = 45 ^ { \circ } , v _ { 0 } = 114$ feet per second
​

A)​ ​
B)​
C)​
D)​
E)​

Question 42

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

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 43

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 $v _ { 0 }$ .​ $\theta = 60 ^ { \circ }$ , $v _ { 0 } = 93$ feet per second
​

A)​
B)​
C)​
D)​
E)​ ​

Question 44

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 $v _ { 0 }$ .​ $\theta = 10 ^ { \circ } , v _ { 0 } = 70$ feet per second
​

A)​
B)​
C)​
D)​
E)​ ​

Question 45

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.​ $\begin{array} { l } x = | t - 1 | \\y = t + 5\end{array}$ ​

A) $x = | y - 1 | \text { OR } y = x + 1 , x \geq 0 \text { and } y = - x + 6 , x \geq 0$
B) $x = | y - 6 | \text { OR } y = x + 6 , x \geq 0 \text { and } y = - x + 6 , x \geq 0$
C) $x = | y + 6 | \text { OR } y = x + 6 , x \geq 0 \text { and } y = x + 6 , x \geq 0$
D) $x = | y - 5 | \text { OR } y = x + 5 , x \geq 0 \text { and } y = - x + 6 , x \geq 0$
E) $x = | y + 5 | \text { OR } y = x - 1 , x \geq 0 \text { and } y = - x + 5 , x \geq 0$

Question 46

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
​

A)​ ​
B)​ ​
C)​ ​
D)​ ​
E)​ ​
​

Question 47

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

A)Involute of circle: $$x = \frac { 1 } { 5 } ( \cos \theta + \Theta \sin \theta ) , y = \frac { 1 } { 5 } ( \sin \theta - \Theta \cos \theta )$$
B)Involute of circle: $x = \frac { 1 } { 5 } ( \cos \theta + \Theta \sin \theta ) , y = \frac { 1 } { 5 } ( \sin \theta + \Theta \cos \theta )$
C)Involute of circle: $x = \frac { 1 } { 5 } ( \cos \Theta - \Theta \sin \theta ) , y = \frac { 1 } { 5 } ( \sin \theta - \Theta \cos \theta )$
D)Involute of circle: $x = \frac { 1 } { 5 } ( \cos \theta - \Theta \sin \theta ) , y = \frac { 1 } { 5 } ( \sin \theta + \Theta \cos \theta )$
E)Involute of circle: $x = \frac { 1 } { 5 } ( \cos \theta + \Theta \sin \theta ) , y = \frac { 1 } { 5 } ( \cos \theta - \Theta \sin \theta )$

Question 48

Select the curve represented by the parametric equations.​ $$\begin{array} { l } x = | t - 2 | \\y = t + 6\end{array}$$ ​

A)​
B)​ ​
C)​
D)​
E)​

Question 49

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.​ $\begin{array} { l } x = 3 \cos \theta \\y = 4 \sin \theta\end{array}$ ​

A) $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 16 } = 1$
B) $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } = 1$
C) $$y = \frac { x } { 4 }$$
D) $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 9 } = 1$
E) $y = \frac { x } { 3 }$

Question 50

Select the curve represented by the parametric equations.​ $$\begin{array} { l } x = 4 ( t + 1 ) \\y = | t - 4 |\end{array}$$ ​

A)​ ​
B)​
C)​
D)​
E)​