Deck 9: Parametric Equations Polar Coordinates and Conic Sections

Question 1

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = t , \quad y = 3 t ^ { 2 } , \quad t \in ( - \infty , \infty )$$

A) $y = x ^ { 2 } , x \in ( - \infty , \infty ) \text {, left to right }$
B) $y = 3 x , x \in ( - \infty , \infty ) \text {, left to right }$
C) $y = 3 x ^ { 2 } , x \in ( - \infty , \infty ) \text {, right to left }$
D) $y = 3 x ^ { 2 } , x \in ( - \infty , \infty ) \text {, left to right }$

Answer 1

$y = 3 x ^ { 2 } , x \in ( - \infty , \infty ) \text {, left to right }$

Question 2

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = 1 - t , \quad y = - 2 t ^ { 2 } , \quad t \geq 0$$

A) $y = - 2 ( x + 1 ) ^ { 2 } , x \in ( - \infty , \infty ) \text {, right to left }$
B) $y = - 2 ( x - 1 ) ^ { 2 } , x \in ( - \infty , \infty ) \text {, right to left }$
C) $y = - 2 ( x - 1 ) ^ { 2 } , x \leq 1 \text {, right to left }$
D) $y = - 2 ( x - 1 ) ^ { 2 } , x \in ( - \infty , \infty ) \text {, left to right }$

Answer 2

$y = - 2 ( x - 1 ) ^ { 2 } , x \leq 1 \text {, right to left }$

Question 3

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = 3 t + 1 , \quad y = 9 t ^ { 2 } - 2 , \quad t \in ( - \infty , \infty )$$

A) $y = ( x - 1 ) ^ { 2 } + 2 , x \in ( - \infty , \infty ) \text {, left to right }$
B) $y = ( x - 1 ) ^ { 2 } - 2 , x \in ( - \infty , \infty ) \text {, right to left }$
C) $y = ( x - 1 ) ^ { 2 } - 2 , x \in ( - \infty , \infty ) \text {, left to right }$
D) $y = ( x - 1 ) ^ { 2 } + 2 , x \in ( - \infty , \infty ) \text {, right to left }$

Answer 3

$y = ( x - 1 ) ^ { 2 } - 2 , x \in ( - \infty , \infty ) \text {, left to right }$

Question 4

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = - 2 t , \quad y = 6 t ^ { 2 } - 1 , \quad t \in [ - 2,3 ]$$

A) $y = \frac { 3 } { 2 } x ^ { 2 } + 1 , x \in ( - \infty , \infty ) \text {, left to right }$
B) $y = \frac { 3 } { 2 } x ^ { 2 } + 1 , x \in ( - \infty , \infty ) \text {, right to left }$
C) $y = \frac { 3 } { 2 } x ^ { 2 } - 1 , x \in ( - \infty , \infty ) \text {, left to right }$
D) $y = \frac { 3 } { 2 } x ^ { 2 } - 1 , - 6 \leq x \leq 4 \text {, right to left }$

Question 5

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = 2 t + 1 , \quad y = 3 t , \quad t \in [ - 1,3 ]$$

A) $y = \frac { 3 } { 2 } x + 1 , \quad - 1 \leq x \leq 7 , \text { left to right }$
B) $y = \frac { 3 } { 2 } x - 1 , \quad - 1 \leq x \leq 7 , \text { left to right }$
C) $y = \frac { 3 } { 2 } ( x - 1 ) , \quad - 1 \leq \mathrm { x } \leq 7 , \text { left to right }$
D) $y = \frac { 3 } { 2 } ( x - 1 ) , \quad - 1 \leq \mathrm { x } \leq 7 , \quad \text { right to left }$

Question 6

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = t - 1 , \quad y = 2 t + 5 , \quad t \in ( - \infty , \infty )$$

A) $y = 2 x - 7 , x \in ( - \infty , \infty ) \text {, left to right }$
B) $y = 2 x - 7 , x \in ( - \infty , \infty ) , \quad \text { right to left }$
C) $y = 2 x + 7 , x \in ( - \infty , \infty ) \text {, left to right }$
D) $y = 2 x + 7 , x \in ( - \infty , \infty ) , \quad \text { right to left }$

Question 7

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = - 4 t , \quad y = - 3 t + 1 , \quad t \in [ - 2,1 ]$$

A) $y = \frac { 3 } { 4 } x + 1 , \quad - 4 \leq x \leq 8 , \quad \text { left to right }$
B) $y = \frac { 3 } { 4 } x - 1 , \quad - 4 \leq x \leq 8 , \quad \text { left to right }$
C) $y = \frac { 3 } { 4 } x - 1 , \quad - 4 \leq x \leq 8 , \quad \text { right to left }$
D) $y = \frac { 3 } { 4 } x + 1 , \quad - 4 \leq x \leq 8 , \quad \text { right to left }$

Question 8

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = t + 1 , \quad y = e ^ { t } , \quad t \in ( - \infty , \infty )$$

A) $y = e ^ { x + 1 } , x \in ( - \infty , \infty ) \text {, left to right }$
B) $y = e ^ { x - 1 } , x \in ( - \infty , \infty ) \text {, left to right }$
C) $y = e ^ { x + 1 } , x \in ( - \infty , \infty ) \text {, right to left }$
D) $y = e ^ { x - 1 } , x \in ( - \infty , \infty ) \text {, right to left }$

Question 9

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = 2 \sin t , \quad y = 2 \cos t , \quad t \in [ 0,2 \pi ]$$

A) $x ^ { 2 } + y ^ { 2 } = 2 , \quad \text { clockwise }$
B) $x ^ { 2 } + y ^ { 2 } = 2 , \quad \text { counterclockwise }$
C) $x ^ { 2 } + y ^ { 2 } = 4 , \quad \text { clockwise }$
D) $x ^ { 2 } + y ^ { 2 } = 4 , \quad \text { counterclockwise }$

Question 10

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = 3 \cos t , \quad y = 3 \sin t , \quad t \in [ 0,2 \pi ]$$

A) $x ^ { 2 } + y ^ { 2 } = 3 , \quad \text { counterclockwise }$
B) $x ^ { 2 } + y ^ { 2 } = 9 , \quad \text { clockwise }$
C) $x ^ { 2 } + y ^ { 2 } = 3 , \quad \text { clockwise }$
D) $x ^ { 2 } + y ^ { 2 } = 9 , \quad \text { counterclockwise }$

Question 11

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = - 2 t - 1 , \quad y = 3 t ^ { 2 } + 2 , \quad t \in ( - \infty , \infty )$$

A) $y = \frac { 3 } { 4 } ( x - 1 ) ^ { 2 } + 2 , \quad x \in ( - \infty , \infty ) , \quad \text { right to left }$
B) $y = \frac { 3 } { 4 } ( x + 1 ) ^ { 2 } - 2 , \quad x \in ( - \infty , \infty ) , \quad \text { right to left }$
C) $y = \frac { 3 } { 4 } ( x - 1 ) ^ { 2 } - 2 , \quad x \in ( - \infty , \infty ) , \quad \text { right to left }$
D) $y = \frac { 3 } { 4 } ( x + 1 ) ^ { 2 } + 2 , \quad x \in ( - \infty , \infty ) , \quad \text { right to left }$

Question 12

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = 3 \sin t , \quad y = 4 \cos t , \quad t \in [ 0,2 \pi ]$$

A) $9 x ^ { 2 } + 9 y ^ { 2 } = 144 , \quad \text { counterclockwise }$
B) $9 x ^ { 2 } + 16 y ^ { 2 } = 144 , \quad \text { counterclockwise }$
C) $16 x ^ { 2 } + 9 y ^ { 2 } = 144 , \quad \text { counterclockwise }$
D) $16 x ^ { 2 } + 9 y ^ { 2 } = 144 , \quad \text { clockwise }$

Question 13

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = 4 \cos t , \quad y = 3 \sin t , \quad t \in [ 0,2 \pi ]$$

A) $16 x ^ { 2 } + 9 y ^ { 2 } = 144 , \quad \text { counterclockwise }$
B) $16 x ^ { 2 } + 9 y ^ { 2 } = 144 , \quad \text { clockwise }$
C) $9 x ^ { 2 } + 16 y ^ { 2 } = 144 , \quad \text { counterclockwise }$
D) $9 x ^ { 2 } + 16 y ^ { 2 } = 144 , \quad \text { clockwise }$

Question 14

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = \sin t , \quad y = \cos 2 t , \quad t \in \left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$$

A) $y = 1 + 2 x ^ { 2 } , \quad - 1 \leq x \leq 1 , \quad \text { left to right }$
B) $y = 1 - 2 x ^ { 2 } , \quad - 1 \leq x \leq 1 , \quad \text { left to right }$
C) $y = 1 + 2 x ^ { 2 } , \quad - 1 \leq x \leq 1 , \quad \text { right to left }$
D) $y = 1 - 2 x ^ { 2 } , \quad - 1 \leq x \leq 1 , \quad \text { right to left }$

Question 15

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = \cos t , \quad y = \cos 2 t , \quad t \in [ 0 , \pi ]$$

A) $y = 2 x ^ { 2 } + 1 , \quad - 1 \leq x \leq 1 , \quad \text { left to right }$
B) $y = 1 + 2 x ^ { 2 } , \quad - 1 \leq x \leq 1 \text {, right to left }$
C) $y = 1 - 2 x ^ { 2 } , \quad - 1 \leq x \leq 1 , \quad \text { right to left }$
D) $y = 2 x ^ { 2 } - 1 , \quad - 1 \leq x \leq 1 , \quad \text { right to left }$

Question 16

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve.x = e^2t, y = t, t ? 0

A) $y = \frac { 1 } { 2 } \ln x , \quad x \geq 1 \text {, right to left }$
B) $y = \frac { 1 } { 2 } \ln x , \quad x \geq 1 \text {, left to right }$
C) $y = 2 \ln x , \quad x \geq 1 , \quad \text { right to left }$
D) $y = 2 \ln x , \quad x \geq 1 , \quad \text { left to right }$

Question 17

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = \cos ^ { 2 } t , \quad y = 3 \sin ^ { 2 } t , \quad t \in \left[ 0 , \frac { \pi } { 2 } \right]$$

A) $y = 3 - 3 x , \quad 0 \leq x \leq 1 \text {, left to right }$
B) $y = 3 + 3 x , \quad 0 \leq x \leq 1 , \quad \text { left to right }$
C) $y = 3 - x , \quad 0 \leq x \leq 1 \text {, left to right }$
D) $y = 3 - 3 x , \quad 0 \leq x \leq 1 \text {, right to left }$

Question 18

Find the equation of the tangent line to the parametric curve: $$x = 2 t + 1 , \quad y = 3 t + 4 , \quad \text { at } t = - 1$$

A) $y = \frac { 3 } { 2 } x - \frac { 5 } { 2 }$
B) $y = \frac { 5 } { 2 } x + \frac { 3 } { 2 }$
C) $y = \frac { 3 } { 2 } x + \frac { 5 } { 2 }$
D) $y = - \frac { 3 } { 2 } x + \frac { 5 } { 2 }$

Question 19

Find the equation of the tangent line to the parametric curve: $$x = t + 1 , \quad y = e ^ { 2 t } , \quad \text { at } t = 0$$

A) y = 2x + 1
B) y = x - 2
C) y = 2x + 3
D) y = 2x - 1

Question 20

Find the equation of the tangent line to the parametric curve: $$x = t ^ { 3 } , \quad y = ( 3 - t ) ^ { 2 } , \quad \text { at } t = 1$$

A) $y = - \frac { 4 } { 3 } x - \frac { 16 } { 3 }$
B) $y = \frac { 4 } { 3 } x + \frac { 16 } { 3 }$
C) $y = - \frac { 4 } { 3 } x + \frac { 16 } { 3 }$
D) $y = \frac { 4 } { 3 } x - \frac { 16 } { 3 }$

Question 21

Find the length of the parametric curve: $$x = e ^ { t } \sin t , \quad y = e ^ { t } \cos t , \quad \mathrm { t } \in [ 0,1 ]$$

A) $\sqrt { 2 } ( e + 1 )$
B) $\sqrt { 3 } ( e - 1 )$
C) $\sqrt { 2 } e$
D) $\sqrt { 2 } ( e - 1 )$

Question 22

Find the length of the parametric curve: $$x = e ^ { 2 t } \cos t , \quad y = e ^ { 2 t } \sin t , \quad \mathrm { t } \in [ 0,1 ]$$

A) $\frac { \sqrt { 5 } } { 2 } ( e - 1 )$
B) $\frac { \sqrt { 5 } } { 2 } \left( e ^ { 2 } + 1 \right)$
C) $\frac { \sqrt { 3 } } { 2 } \left( e ^ { 2 } - 1 \right)$
D) $\frac { \sqrt { 5 } } { 2 } \left( e ^ { 2 } - 1 \right)$

Question 23

Find the length of the parametric curve: $$x = t ^ { 2 } , \quad y = 3 - t ^ { 3 } , \quad \mathrm { t } \in [ 0,2 ]$$

Question 24

Find the length of the parametric curve: $$x = 2 \sin ^ { 2 } t , \quad y = 3 \cos ^ { 2 } t , \quad \mathrm { t } \in \left[ 0 , \frac { 3 \pi } { 2 } \right]$$

A) $\frac { \sqrt { 26 } } { 2 }$
B) $\frac { \sqrt { 52 } } { 4 }$
C) $\frac { \sqrt { 52 } } { 2 }$
D) $\frac { \sqrt { 54 } } { 2 }$

Question 25

Find the rectangular coordinates for the following points whose polar coordinates are given:
A .(3, $\pi$ /6)

B. (-3, $\pi$ /2)
C. (3, 3 $\pi$ /2)

D. (3, - $\pi$ /6)

Question 26

Find the rectangular coordinates for the following points whose polar coordinates are given:
A .(2, - $\pi$ /2)
B.(2, -3 $\pi$ /2)
C .(2, $\pi$ /3)
D .(2, $\pi$ )

Question 27

Find the polar coordinates of the point (4, 4 $\sqrt { 3 }$ ) given in rectangular coordinates for $\theta \in [ 0,2 \pi ]$ and $r \geq 0$

A) (4, $\pi$ /3)
B) (8, $\pi$ /6)
C) (8, $\pi$ /3)
D) (4, $\pi$ /6)

Question 28

Find the polar coordinates of the point (0, -2) given in rectangular coordinates for $\theta \in [ 0,2 \pi ]$ and $r \geq 0$

A) $\left( 2 , \frac { \pi } { 2 } \right)$
B) $\left( - 2 , \frac { 3 \pi } { 2 } \right)$
C) $\left( 2 , \frac { \pi } { 3 } \right)$
D) $\left( 2 , \frac { 3 \pi } { 2 } \right)$

Question 29

Find the polar coordinates of the point (-3, 0) given in rectangular coordinates for $\theta \in [ 0,2 \pi ]$ and $r \geq 0$

A) (3, $\pi$ /2)
B) (3, 2 $\pi$ )
C) (3, $\pi$ )
D) (3, 3 $\pi$ /2)

Question 30

Find the polar coordinates of the point (2, -2) given in rectangular coordinates for $\theta \in [ 0,2 \pi ]$ and $r \geq 0$

Question 31

Find the polar coordinates of the point $( - 2 \sqrt { 3 } , 2 )$ given in rectangular coordinates for $\theta \in [ 0,2 \pi ]$ and $r \geq 0$

A) $\left( 4 , \frac { \pi } { 6 } \right)$
B) $\left( 4 , \frac { \pi } { 3 } \right)$
C) $\left( 4 , \frac { 2 \pi } { 3 } \right)$
D) $\left( 4 , \frac { 5 \pi } { 6 } \right)$

Question 32

Convert the equation $\theta = \frac { \pi } { 3 }$ to rectangular coordinates.

A) $y = \frac { 1 } { \sqrt { 3 } } x$
B) $y = \sqrt { 3 } x$
C) $y = \frac { \sqrt { 2 } } { 2 } x$
D) y = x

Question 33

Convert the equation $r = 3 \cos \theta$ to rectangular coordinates.

A) $x ^ { 2 } + 3 x + y ^ { 2 } = 0$
B) $x ^ { 2 } + y ^ { 2 } - 3 y = 0$
C) $x ^ { 2 } + y ^ { 2 } + 3 y = 0$
D) $x ^ { 2 } - 3 x + y ^ { 2 } = 0$

Question 34

Convert the equation $r = 2 \sin \theta$ to rectangular coordinates.

A) $x ^ { 2 } - 2 x + y ^ { 2 } = 0$
B) $x ^ { 2 } + y ^ { 2 } + 2 y = 0$
C) $x ^ { 2 } + y ^ { 2 } - 2 y = 0$
D) $x ^ { 2 } + 2 x + y ^ { 2 } = 0$

Question 35

Convert the equation $r = \sin 2 \theta$ to rectangular coordinates.

Question 36

Convert the equation $r ^ { 2 } = \cos \theta$ to rectangular coordinates.

Question 37

Convert the equation $r = 5 \sec \theta$ to rectangular coordinates.

A) y = 5
B) y = 3
C) $\theta$ = $\pi$ /4
D) x = 5

Question 38

Convert the equation $r = - 2 \csc \theta$ to rectangular coordinates.

A) y = 2
B) x = 2
C) y = -2
D) x = -2

Question 39

Convert the equation x = -2 to polar coordinates.

A) $r = 2 \sec \theta$
B) $r = - 2 \csc \theta$
C) $r = 2 \csc \theta$
D) $r = - 2 \sec \theta$

Question 40

Convert the equation y = 4 to polar coordinates.

A) $r = 4 \sec \theta$
B) $r = - 4 \sec \theta$
C) $r = 4 \csc \theta$
D) $r = 4 \cos \theta$

Question 41

Convert the equation y = x to polar coordinates.

A) $\theta = \frac { \pi } { 3 }$
B) $\theta = \frac { \pi } { 6 }$
C) $\theta = \frac { \pi } { 2 }$
D) $\theta = \frac { \pi } { 4 }$

Question 42

Convert the equation y = x + 2 to polar coordinates.

Question 43

Find all the points of intersection of the curves, for $0 \leq \theta < 2 \pi$ : $r = \sin \theta \text { and } r = \cos \theta$

Question 44

Find all the points of intersection of the curves for $0 \leq \theta < 2 \pi$ : $r = 1 + \sin \theta \text { and } r = 1 - \sin \theta$

Question 45

Find all the points of intersection of the curves for $0 \leq \theta < 2 \pi$ : $r = 1 - \sin \theta \text { and } r = 1 + \cos \theta$

Question 46

Find all the points of intersection of the curves for $0 \leq \theta < 2 \pi$ : $r = 2 \sin 2 \theta \text { and } r = 1$

Question 47

Find a definite integral expression that represents the area of the region inside one loop of the lemniscate $r ^ { 2 } = \sin 2 \theta$ Then find the exact value of the integral.

Question 48

Find a definite integral expression that represents the area of the region inside both the cardioid $r = 1 + \sin \theta$ and outside the cardioid $r = 1 - \sin \theta$ Then find the exact value of the integral.

Question 49

Find a definite integral expression that represents the area of the region inside $r = 1 + \cos \theta$ Then find the exact value of the integral.

Question 50

Find a definite integral expression that represents the area of the region inside one petal of the polar rose $r = 2 \cos 3 \theta$ Then find the exact value of the integral.

Question 51

Find a definite integral expression that represents the area of the region inside one petal of the polar rose $r = 3 \sin 2 \theta$ Then find the exact value of the integral.

Question 52

Find a definite integral expression that represents the area of the region inside the circle $r = 3 \sin \theta$ and outside the cardioid $r = 1 + \sin \theta$ Then find the exact value of the integral.

Question 53

Find a definite integral expression that represents the area of the region between the loops of the limacon $r = 1 + 2 \cos \theta$ Then find the exact value of the integral.

Question 54

Describe the conic: $x ^ { 2 } + 4 x + y ^ { 2 } - 8 y - 24 = 0$

Question 55

Describe the conic: $x ^ { 2 } - 6 x + 4 y - 3 = 0$

Question 56

Describe the conic: $4 x ^ { 2 } - 8 x + 2 y ^ { 2 } - 12 y - 3 = 0$

Question 57

Write the equation of the ellipse with foci $( 0 , \pm 3 )$ and length of minor axis 8.

Question 58

Write the equation of the ellipse with foci $( \pm 2,0 )$ and length of major axis 6.

Question 59

Write the equation of the ellipse with foci (-1, 1) and (-1, 3) and length of minor axis 4.

Question 60

Find the equation of the hyperbola with foci ( $\pm 3$ , 0) and vertices $( \pm 2,0 )$

Question 61

Find the equation of the hyperbola with foci $( \pm 3,0 )$ and asymptotes y = $\pm 2 x$

Question 62

Find the equation of a parabola with focus (-2, 0) and vertex (0, 0).

Question 63

Find the equation of a parabola with focus (0, 4) and vertex (0, 0).