Exam 9: Parametric Equations Polar Coordinates and Conic Sections

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. $x = 3 \sin t , \quad y = 4 \cos t , \quad t \in [ 0,2 \pi ]$

(Multiple Choice)

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Question 41

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

(Essay)

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Question 42

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

(Multiple Choice)

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Question 43

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, $\pi$ )

(Essay)

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Question 44

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

(Essay)

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Question 45

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. $x = 2 t + 1 , \quad y = 3 t , \quad t \in [ - 1,3 ]$

(Multiple Choice)

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Question 46

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

(Multiple Choice)

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Question 47

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

(Essay)

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Question 48

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. $x = \cos ^ { 2 } t , \quad y = 3 \sin ^ { 2 } t , \quad t \in \left[ 0 , \frac { \pi } { 2 } \right]$

(Multiple Choice)

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Question 49

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

(Essay)

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Question 50

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. $x = - 4 t , \quad y = - 3 t + 1 , \quad t \in [ - 2,1 ]$

(Multiple Choice)

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Question 51

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

(Multiple Choice)

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Question 52

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

(Essay)

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Question 53

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 = e²^t, y = t, t ? 0

(Multiple Choice)

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Question 54

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

(Multiple Choice)

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Question 55

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. $x = t + 1 , \quad y = e ^ { t } , \quad t \in ( - \infty , \infty )$

(Multiple Choice)

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Question 56

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. $x = 3 t + 1 , \quad y = 9 t ^ { 2 } - 2 , \quad t \in ( - \infty , \infty )$

(Multiple Choice)

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Question 57

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.

(Essay)

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Question 58

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

(Multiple Choice)

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Question 59

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. $x = t , \quad y = 3 t ^ { 2 } , \quad t \in ( - \infty , \infty )$

(Multiple Choice)

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Question 60

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. $x = 3 \sin t , \quad y = 4 \cos t , \quad t \in [ 0,2 \pi ]$

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

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

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, $\pi$ )

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

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. $x = 2 t + 1 , \quad y = 3 t , \quad t \in [ - 1,3 ]$

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

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

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. $x = \cos ^ { 2 } t , \quad y = 3 \sin ^ { 2 } t , \quad t \in \left[ 0 , \frac { \pi } { 2 } \right]$

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

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. $x = - 4 t , \quad y = - 3 t + 1 , \quad t \in [ - 2,1 ]$

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

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

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 = e²^t, y = t, t ? 0

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

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. $x = t + 1 , \quad y = e ^ { t } , \quad t \in ( - \infty , \infty )$

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. $x = 3 t + 1 , \quad y = 9 t ^ { 2 } - 2 , \quad t \in ( - \infty , \infty )$

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.

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

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. $x = t , \quad y = 3 t ^ { 2 } , \quad t \in ( - \infty , \infty )$

Limits

Derivatives

Applications of the Derivative

Definite Integrals

Techniques of Integration

Applications of Integration

Sequences and Series

Power Series

Vectors

Vector Functions

Multivariable Functions

Double and Triple Integrals

Vector Analysis

Functions and Precalculus

Filters

Exam 9: Parametric Equations Polar Coordinates and Conic Sections

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=3sin⁡t,y=4cos⁡t,t∈[0,2π]x = 3 \sin t , \quad y = 4 \cos t , \quad t \in [ 0,2 \pi ]x=3sint,y=4cost,t∈[0,2π]

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

Convert the equation r=3cos⁡θr = 3 \cos \thetar=3cosθ to rectangular coordinates.

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

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

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=2t+1,y=3t,t∈[−1,3]x = 2 t + 1 , \quad y = 3 t , \quad t \in [ - 1,3 ]x=2t+1,y=3t,t∈[−1,3]

Find the length of the parametric curve: x=2sin⁡2t,y=3cos⁡2t,t∈[0,3π2]x = 2 \sin ^ { 2 } t , \quad y = 3 \cos ^ { 2 } t , \quad \mathrm { t } \in \left[ 0 , \frac { 3 \pi } { 2 } \right]x=2sin2t,y=3cos2t,t∈[0,23π​]

Find all the points of intersection of the curves for 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π : r=1+sin⁡θ and r=1−sin⁡θr = 1 + \sin \theta \text { and } r = 1 - \sin \thetar=1+sinθ and r=1−sinθ

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

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=−4t,y=−3t+1,t∈[−2,1]x = - 4 t , \quad y = - 3 t + 1 , \quad t \in [ - 2,1 ]x=−4t,y=−3t+1,t∈[−2,1]

Convert the equation r=−2csc⁡θr = - 2 \csc \thetar=−2cscθ to rectangular coordinates.

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

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 = e2t, y = t, t ? 0

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

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,y=et,t∈(−∞,∞)x = t + 1 , \quad y = e ^ { t } , \quad t \in ( - \infty , \infty )x=t+1,y=et,t∈(−∞,∞)

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=3t+1,y=9t2−2,t∈(−∞,∞)x = 3 t + 1 , \quad y = 9 t ^ { 2 } - 2 , \quad t \in ( - \infty , \infty )x=3t+1,y=9t2−2,t∈(−∞,∞)

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

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

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,y=3t2,t∈(−∞,∞)x = t , \quad y = 3 t ^ { 2 } , \quad t \in ( - \infty , \infty )x=t,y=3t2,t∈(−∞,∞)

Limits

Derivatives

Applications of the Derivative

Definite Integrals

Techniques of Integration

Applications of Integration

Sequences and Series

Power Series

Vectors

Vector Functions

Multivariable Functions

Double and Triple Integrals

Vector Analysis

Functions and Precalculus

Filters

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. $x = 3 \sin t , \quad y = 4 \cos t , \quad t \in [ 0,2 \pi ]$

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

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, $\pi$ )

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. $x = 2 t + 1 , \quad y = 3 t , \quad t \in [ - 1,3 ]$

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

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

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. $x = - 4 t , \quad y = - 3 t + 1 , \quad t \in [ - 2,1 ]$

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

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

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 = e²^t, y = t, t ? 0

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

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. $x = t + 1 , \quad y = e ^ { t } , \quad t \in ( - \infty , \infty )$

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. $x = 3 t + 1 , \quad y = 9 t ^ { 2 } - 2 , \quad t \in ( - \infty , \infty )$

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.

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

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. $x = t , \quad y = 3 t ^ { 2 } , \quad t \in ( - \infty , \infty )$