Exam 12: Parametric Equations and Polar Coordinates

If the equation represents a hyperbola, find the center, foci, and asymptotes. If the equation represents an ellipse, find the center, vertices, and foci. If the equation represents a circle, find the center and radius. If the equation represents a parabola, find the focus and directrix. - $y ^ { 2 } - x ^ { 2 } - 14 y + 12 x - 36 = 0$

(Multiple Choice)

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Question 241

Find the eccentricity of the hyperbola. - $81 y ^ { 2 } - 16 x ^ { 2 } = 1296$

(Multiple Choice)

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Question 242

Solve the problem. -Find the foci and asymptotes of the following hyperbola: $4 y ^ { 2 } - 25 x ^ { 2 } = 100$

(Multiple Choice)

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Question 243

Solve the problem. -Find the vertices and foci of the following hyperbola: $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 91 } = 1$

(Multiple Choice)

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Question 244

Determine if the given polar coordinates represent the same point. - $( 9 , \pi / 4 ) , ( 9,5 \pi / 4 )$

(True/False)

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Question 245

Find the slope of the polar curve at the indicated point. - $\mathrm { r } = 9 , \theta = \frac { \pi } { 2 }$

(Multiple Choice)

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Question 246

Determine the symmetries of the curve. - $r ^ { 2 } = 2 \sin 4 \theta$

(Multiple Choice)

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Question 247

Solve the problem. -The ellipse $\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 16 } = 1$ is shifted down 3 units and right 8 units. Find an equation for the new ellipse and find the new center.

(Multiple Choice)

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Question 248

Find the eccentricity of the ellipse. - $9 x ^ { 2 } + 4 y ^ { 2 } = 36$

(Multiple Choice)

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Question 249

Find the directrices of the ellipse. - $4 \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } = 4$

(Multiple Choice)

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Question 250

$\text { Find the polar coordinates, } 0 \leq \theta < 2 \pi \text { and } r \geq 0 \text {, of the point given in Cartesian coordinates. }$ -(-2, 0)

(Multiple Choice)

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Question 251

Replace the Cartesian equation with an equivalent polar equation. -y = 10

(Multiple Choice)

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Question 252

Graph the polar equation. - $r = 8 \cos 6 \theta$ $Graph the polar equation. - r = 8 \cos 6 \theta$

(Multiple Choice)

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Question 253

Determine if the given polar coordinates represent the same point. - $( 6 , \pi / 6 ) , ( 6,7 \pi / 6 )$

(True/False)

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Question 254

Find the area of the specified region. -Inside the outer loop and outside the inner loop of the limacon $r = 8 \cos \theta - 4$

(Multiple Choice)

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Question 255

Parametric equations and and a parameter interval for the motion of a particle in the xy-plane are given. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction of motion. - $x=2 \cos t, y=2 \sin t, \pi \leq t \leq 2 \pi$ $Parametric equations and and a parameter interval for the motion of a particle in the xy-plane are given. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction of motion. - x=2 \cos t, y=2 \sin t, \pi \leq t \leq 2 \pi$

(Multiple Choice)

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Question 256

Graph. - $y^{2}-x^{2}=\frac{1}{4}$ $Graph. - y^{2}-x^{2}=\frac{1}{4}$

(Multiple Choice)

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Question 257

Solve the problem. -The ellipse $\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 11 } = 1$ is shifted down 6 units and right 4 units. Find an equation for the new ellipse and find the new vertices.

(Multiple Choice)

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Question 258

Graph the parabola. - $y=x^{2}$ $Graph the parabola. - y=x^{2}$

(Multiple Choice)

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Question 259

Graph the pair of parametric equations with the aid of a graphing calculator. - $x=\cos 5 t, y=\sin 4 t, 0 \leq t \leq 2 \pi$ $Graph the pair of parametric equations with the aid of a graphing calculator. - x=\cos 5 t, y=\sin 4 t, 0 \leq t \leq 2 \pi$

(Multiple Choice)

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Question 260

Showing 241 - 260 of 396

Find the eccentricity of the hyperbola. - $81 y ^ { 2 } - 16 x ^ { 2 } = 1296$

Solve the problem. -Find the foci and asymptotes of the following hyperbola: $4 y ^ { 2 } - 25 x ^ { 2 } = 100$

Solve the problem. -Find the vertices and foci of the following hyperbola: $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 91 } = 1$

Determine if the given polar coordinates represent the same point. - $( 9 , \pi / 4 ) , ( 9,5 \pi / 4 )$

Find the slope of the polar curve at the indicated point. - $\mathrm { r } = 9 , \theta = \frac { \pi } { 2 }$

Determine the symmetries of the curve. - $r ^ { 2 } = 2 \sin 4 \theta$

Solve the problem. -The ellipse $\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 16 } = 1$ is shifted down 3 units and right 8 units. Find an equation for the new ellipse and find the new center.

Find the eccentricity of the ellipse. - $9 x ^ { 2 } + 4 y ^ { 2 } = 36$

Find the directrices of the ellipse. - $4 \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } = 4$

$\text { Find the polar coordinates, } 0 \leq \theta < 2 \pi \text { and } r \geq 0 \text {, of the point given in Cartesian coordinates. }$ -(-2, 0)

Replace the Cartesian equation with an equivalent polar equation. -y = 10

Graph the polar equation. - $r = 8 \cos 6 \theta$ $Graph the polar equation. - r = 8 \cos 6 \theta$

Determine if the given polar coordinates represent the same point. - $( 6 , \pi / 6 ) , ( 6,7 \pi / 6 )$

Find the area of the specified region. -Inside the outer loop and outside the inner loop of the limacon $r = 8 \cos \theta - 4$

Graph. - $y^{2}-x^{2}=\frac{1}{4}$ $Graph. - y^{2}-x^{2}=\frac{1}{4}$

Solve the problem. -The ellipse $\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 11 } = 1$ is shifted down 6 units and right 4 units. Find an equation for the new ellipse and find the new vertices.

Graph the parabola. - $y=x^{2}$ $Graph the parabola. - y=x^{2}$

Graph the pair of parametric equations with the aid of a graphing calculator. - $x=\cos 5 t, y=\sin 4 t, 0 \leq t \leq 2 \pi$ $Graph the pair of parametric equations with the aid of a graphing calculator. - x=\cos 5 t, y=\sin 4 t, 0 \leq t \leq 2 \pi$

Functions

Limits and Continuity

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Applications of Derivatives

Integrals

Applications of Definite Integrals

Integrals and Transcendental Functions

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First-Order Differential Equations

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector-Valued Functions and Motion in Space

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Exam 12: Parametric Equations and Polar Coordinates

Find the eccentricity of the hyperbola. - 81y2−16x2=129681 y ^ { 2 } - 16 x ^ { 2 } = 129681y2−16x2=1296

Solve the problem. -Find the foci and asymptotes of the following hyperbola: 4y2−25x2=1004 y ^ { 2 } - 25 x ^ { 2 } = 1004y2−25x2=100

Solve the problem. -Find the vertices and foci of the following hyperbola: x29−y291=1\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 91 } = 19x2​−91y2​=1

Determine if the given polar coordinates represent the same point. - (9,π/4),(9,5π/4)( 9 , \pi / 4 ) , ( 9,5 \pi / 4 )(9,π/4),(9,5π/4)

Find the slope of the polar curve at the indicated point. - r=9,θ=π2\mathrm { r } = 9 , \theta = \frac { \pi } { 2 }r=9,θ=2π​

Determine the symmetries of the curve. - r2=2sin⁡4θr ^ { 2 } = 2 \sin 4 \thetar2=2sin4θ

Solve the problem. -The ellipse x25+y216=1\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 16 } = 15x2​+16y2​=1 is shifted down 3 units and right 8 units. Find an equation for the new ellipse and find the new center.

Find the eccentricity of the ellipse. - 9x2+4y2=369 x ^ { 2 } + 4 y ^ { 2 } = 369x2+4y2=36

Find the directrices of the ellipse. - 4x2+y2=44 \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } = 44x2+y2=4

Replace the Cartesian equation with an equivalent polar equation. -y = 10

Graph the polar equation. - r=8cos⁡6θr = 8 \cos 6 \thetar=8cos6θ

Determine if the given polar coordinates represent the same point. - (6,π/6),(6,7π/6)( 6 , \pi / 6 ) , ( 6,7 \pi / 6 )(6,π/6),(6,7π/6)

Find the area of the specified region. -Inside the outer loop and outside the inner loop of the limacon r=8cos⁡θ−4r = 8 \cos \theta - 4r=8cosθ−4

Graph. - y2−x2=14y^{2}-x^{2}=\frac{1}{4}y2−x2=41​

Solve the problem. -The ellipse x25+y211=1\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 11 } = 15x2​+11y2​=1 is shifted down 6 units and right 4 units. Find an equation for the new ellipse and find the new vertices.

Graph the parabola. - y=x2y=x^{2}y=x2

Graph the pair of parametric equations with the aid of a graphing calculator. - x=cos⁡5t,y=sin⁡4t,0≤t≤2πx=\cos 5 t, y=\sin 4 t, 0 \leq t \leq 2 \pix=cos5t,y=sin4t,0≤t≤2π

Functions

Limits and Continuity

Derivatives

Applications of Derivatives

Integrals

Applications of Definite Integrals

Integrals and Transcendental Functions

Techniques of Integration

First-Order Differential Equations

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector-Valued Functions and Motion in Space

Partial Derivatives

Multiple Integrals

Integrals and Vector Fields

Filters

Find the eccentricity of the hyperbola. - $81 y ^ { 2 } - 16 x ^ { 2 } = 1296$

Solve the problem. -Find the foci and asymptotes of the following hyperbola: $4 y ^ { 2 } - 25 x ^ { 2 } = 100$

Solve the problem. -Find the vertices and foci of the following hyperbola: $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 91 } = 1$

Determine if the given polar coordinates represent the same point. - $( 9 , \pi / 4 ) , ( 9,5 \pi / 4 )$

Find the slope of the polar curve at the indicated point. - $\mathrm { r } = 9 , \theta = \frac { \pi } { 2 }$

Determine the symmetries of the curve. - $r ^ { 2 } = 2 \sin 4 \theta$

Solve the problem. -The ellipse $\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 16 } = 1$ is shifted down 3 units and right 8 units. Find an equation for the new ellipse and find the new center.

Find the eccentricity of the ellipse. - $9 x ^ { 2 } + 4 y ^ { 2 } = 36$

Find the directrices of the ellipse. - $4 \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } = 4$

Graph the polar equation. - $r = 8 \cos 6 \theta$ $Graph the polar equation. - r = 8 \cos 6 \theta$

Determine if the given polar coordinates represent the same point. - $( 6 , \pi / 6 ) , ( 6,7 \pi / 6 )$

Find the area of the specified region. -Inside the outer loop and outside the inner loop of the limacon $r = 8 \cos \theta - 4$

Graph. - $y^{2}-x^{2}=\frac{1}{4}$ $Graph. - y^{2}-x^{2}=\frac{1}{4}$

Solve the problem. -The ellipse $\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 11 } = 1$ is shifted down 6 units and right 4 units. Find an equation for the new ellipse and find the new vertices.

Graph the parabola. - $y=x^{2}$ $Graph the parabola. - y=x^{2}$

Graph the pair of parametric equations with the aid of a graphing calculator. - $x=\cos 5 t, y=\sin 4 t, 0 \leq t \leq 2 \pi$ $Graph the pair of parametric equations with the aid of a graphing calculator. - x=\cos 5 t, y=\sin 4 t, 0 \leq t \leq 2 \pi$