Deck 9: Conics, Parametric Curves, and Polar Curves

Question 1

Find the equation of the parabola passing through the point (2, 5), having vertex at the origin and axis of symmetry along the y-axis.

A) 5 = y
B) 5 = 4y
C) 4 = 5y
D) 3 = 4y
E) = 3y

Answer 1

5 = 4y

Question 2

For the parabola + 6x + 4y + 5 = 0, find the vertex, the focus, and the directrix.

A) Vertex (3, 1), Focus (3, 2), Directrix y = 0
B) Vertex (3, 1), Focus (3, 0), Directrix y = 2
C) Vertex (-3, 1), Focus (-3, 2), Directrix y = 0
D) Vertex (-3, 1), Focus (-3, 0), Directrix y = 2
E) Vertex (-3, -1), Focus (-3, 0), Directrix y = 2

Answer 2

Vertex (-3, 1), Focus (-3, 0), Directrix y = 2

Question 3

Find an equation of an ellipse satisfying the given conditions: Foci (-3, 0) and (3, 0) and length of major axis 6.

A) + = 1
B) + = 1
C) + = 1
D) + = 1
E) + = 1

Answer 3

+ = 1

Question 4

Find an equation of a parabola satisfying the given conditions Focus (4, 1) and directrixx = -2.

A) = 12x
B) = 12(x - 1)
C) = 12(x + 1)
D) = -12(x - 1)
E) = 12x + 1

Question 5

Find an equation of a parabola satisfying the given conditions: Focus (2, 0) and directrix y = 2?.

A) = 4 $\pi$ (1 - y)
B) = 4( $\pi$ - y)
C) = 4 $\pi$ ( $\pi$ - y)
D) = 4 $\pi$ ( $\pi$ - y)
E) = 4 $\pi$ ( $\pi$ - y)

Question 6

Find the centre, eccentricity, and foci of the ellipse + = 1.

A) Centre (1, -3); = ; foci (1 ± , -3)
B) Centre (-1, 3); 11ee7b11_3aa8_6932_ae82_3b68ee6209d9_TB9661_11 = ; foci (-1 ± , 3)
C) Centre (1, 3); 11ee7b11_3aa8_6932_ae82_3b68ee6209d9_TB9661_11 = ; foci (1 ± , 3)
D) Centre (-1, -3); 11ee7b11_3aa8_6932_ae82_3b68ee6209d9_TB9661_11 = ; foci (-1 ± , -3)
E) Centre (1, -3); 11ee7b11_3aa8_6932_ae82_3b68ee6209d9_TB9661_11 = ; foci (1 ± , -3)

Question 7

Find all values of the constant real number k so that the second degree equation represents a pair of lines.

A) k = -1, k =
B) k = 1, -
C) k = - ,
D) - $\infty$ < k < $\infty$
E) k $\neq$ 0

Question 8

Find an equation of an ellipse containing the point (- , ) and with vertices (0, -3) and (0, 3).

A) + = 1
B) - = 1
C) + = 1
D) + = 1
E) + = 1

Question 9

For the hyperbola - = 8x - 2y - 13, find the centre, the vertices, the foci, and the asymptotes.

A) Centre (4, 1), Vertices (4 ± , 1), Foci (4 ± 2, 1), Asymptotes x - y = 3 and x + y = 5
B) Centre (-4, -1), Vertices (-4 ± , -1), Foci (-4 ± 2, -1), Asymptotes x - y = -3 and x + y = -5
C) Centre (4, 1), Vertices (4 ±2 , 1), Foci (4 ± 2, 1), Asymptotes x - y = 3 and x + y = 5
D) Centre (-4, 1), Vertices (-4 ±2 , 1), Foci (-4 ± 2, 1), Asymptotes x - y = -3 and x + y = 5
E) Centre (4, -1), Vertices (4 ± , -1), Foci (4 ± 2, -1), Asymptotes x + y = 3 and x - y = 5

Question 10

The maximum distance of the Earth from the sun is 9.3 × kilometres. The minimum distance is kilometres. The sun is at one focus of the elliptical orbit. Find the distance from the sun to the other focus.

A) 3.2 × 10⁶ kilometres
B) 4.8 × 10⁶ kilometres
C) 6.4 × 10⁶ kilometres
D) 1.6 × 10⁶ kilometres
E) 8.0 × 10⁶ kilometres

Question 11

To eliminate the xy-term from the general equation of a conic section, , , we rotate the coordinate axes about the origin through an , where cot(2 $\theta$ ) = .

Question 12

Find the centre, the foci, and the asymptotes of the hyperbola 4 - 9 -16x - 54y = 101.

A) Centre (2, -3), Foci (2 ± , -3), Asymptotes = ±
B) Centre (-2, 3), Foci (-2 ± , 3), Asymptotes = ±
C) Centre (2, -3), Foci (2 ± , -3), Asymptotes = ±
D) Centre (2, 3), Foci (2 ± , 3), Asymptotes = ±
E) Centre (-2, 3), Foci (-2 ± , 3), Asymptotes = ±

Question 13

Find an equation of a hyperbola with vertices (3, 7) and (-3, 7) and = .

A) - = 1
B) - = 1
C) - = 1
D) - = 1
E) - = 1

Question 14

Find an equation of a hyperbola with vertices (-1, 3) and (-1, 7) and = 4.

A) - = 1
B) - = 1
C) - = 1
D) - = 1
E) - = 1

Question 15

Find the angle at which the parabolas y² = 4x + 4 and y² = -6x + 9 intersect at each of their intersection points.

A) 90º at each intersection point
B) 120º at each intersection point
C) 60º at each intersection point
D) 75º at each intersection point
E) 50º at each intersection point

Question 16

Find the points on the hyperbola x² - y² = 1 nearest to the point (0, 1).

A) ,
B) ,
C) ,
D) ,
E) ,

Question 17

A circle passes through both foci of an ellipse and is tangent to the ellipse at two points. Find the eccentricity of the ellipse.

A)
B)
C)
D)
E)

Question 18

A conic section is given by the equation 4x² + 10xy + 4y² = 36.Use rotation of coordinate axes through an appropriate acute angle $\theta$ to find the new equation of the conic section in the uv-coordinate axes , where x = u cos( $\theta$ ) - v sin( $\theta$ ) , y = u sin( $\theta$ ) + v cos( $\theta$ ). Then identify the conic section.

A) + = 1, an ellipse
B) + = 4, a circle
C) - = 1, a hyperbola
D) + = 1, an ellipse
E) - = 1, a hyperbola

Question 19

Find the equation to the ellipse for which (1, -1) is a focus, x - y = 3 is the corresponding directrix, and the eccentricity is 1/2.

A) 3x² - 2xy + 3y² - 2x + 2y - 1 = 0
B) 3x² + 2xy + 3y² - 2x + 2y - 1 = 0
C) 7x² - 2xy + 7y² - 10x + 10y + 7 = 0
D) 7x² + 2xy + 7y² - 10x + 10y + 7 = 0
E) 7x² + 2xy + 7y² - 5x + 5y - 2 = 0

Question 20

Find the equation of the parabola whose focus is (2, -1) and directrix is x + 2y -1 = 0.

A) 4x² - 4xy + y² -18x + 14y + 24 = 0
B) 5x² - 4xy + y² -18x + 14y + 24 = 0
C) x² - 4xy + 4y² -18x + 14y + 24 = 0
D) x² - 4xy + 5y² -18x + 14y + 24 = 0
E) 4x² - 4xy + 4y² -18x + 14y + 24 = 0

Question 21

Which of the following sets of parametric equations constitute a parametrization of the whole parabola y = x²?

A) (a), (c), and (e) only
B) (a) and (e) only
C) (a), (b), and (c) only
D) all of them
E) none of them

Question 22

What do the parametric equations x = 7 cos(t) and y = 3 sin(t) describe?

A) ellipse
B) hyperbola
C) circle
D) parabola
E) line

Question 23

What do the parametric equations x = t² + 3t and y = t + 4 describe?

A) parabola that opens to the right
B) parabola that opens to the left
C) ellipse
D) hyperbola
E) line

Question 24

A plane curve C is given parametrically by the functions x (t) = cosh(t) - 2, y(t) = sinh(t), t R.
A Cartesian equation of the curve C is given by:

A) - = 1
B) + = 1
C) - =1
D) + = 5
E) - = 5

Question 25

Find parametric equations of the plane curve C given by 4x² + 9y² - 8x -32 = 0.

A) x(t) = 1 + 2cos(t), y(t) = 3sin(t), t [0 , 2 $\pi$ ]
B) x(t) = - 1 + 3cos(t), y(t) = 2sin(t), t11ee7b17_3372_5854_ae82_d19d2ea0c252_TB9661_11 [0 , 2 $\pi$ ]
C) x(t) = - 1 + 2cos(t), y(t) = 3sin(t), t11ee7b17_3372_5854_ae82_d19d2ea0c252_TB9661_11 [0 , 2 $\pi$ ]
D) x(t) = 1 + 3cos(t), y(t) = 2sin(t), t 11ee7b17_3372_5854_ae82_d19d2ea0c252_TB9661_11 [0 , 2 $\pi$ ]
E)

Question 26

Find the Cartesian coordinates of points of intersection of the plane parametric curves , and x = , y = -u - 1.

Question 27

A plane curve C is given parametrically by x = tan(t) - 2, y = sec(t), t (- , ).Find the Cartesian equation of the curve C.

A) - = 1, y $\ge$ 1
B) - = 1, y $\ge$ 1
C) + = 1, - $\infty$ < y < $\infty$
D) - = 5, -1 $\le$ y $\le$ 1
E) - = 1, y $\le$ - 1

Question 28

The equations x(t) = , y(t) = , -1 $\le$ t $\le$ 1 are the parametric equations of

A) the whole circle centred at (0 , 0) and is of radius 1 unit
B) the left half of the circle centred at (0 , 0) and is of radius 1 unit
C) the bottom half of the circle centred at (0 , 0) and is of radius 1 unit
D) the top half of the circle centred at (0 , 0) and is of radius 1 unit
E) the right half of the circle centred at (0 , 0) and is of radius 1 unit

Question 29

Describe the curve x = 3 - cos(t), y = -2 + 2 sin(t).

A) ellipse, centre (3, -2) with major axis along the line x = 3
B) ellipse, centre (3, -2) with major axis along the line y = -2
C) hyperbola, centre (3, -2) with transverse axis along the line x = -3
D) hyperbola, centre (3, -2) with transverse axis along the line x = 3
E) ellipse, centre (-3, 2) with major axis along the line x = -3

Question 30

Parametrize the curve y = + 3x using its slope m as the parameter.

A) x = , y =
B) x = , y =
C) x = , y =
D) x = , y =
E) x = , y =

Question 31

Which of the following plane parametric curves is a parametrization of an ellipse centred at (4, -2)?

A) x = 3 - 4cos(t), y = 5 + 2sin(t), 0 $\le$ t $\le$ 2 $\pi$
B) x = 3 + 4cos(t), y = 5 - 2sin(t), 0 $\le$ t $\le$ 2 $\pi$
C) x = 4 + 3cos(t), y = -2 + 5sin(t), 0 $\le$ t $\le$ 2 $\pi$
D) x = - 4 + 3cos(t), y = 2 + 5sin(t), 0 $\le$ t $\le$ 2 $\pi$
E) x = 4 - 2cos(t), y = 4 - 2sin(t), 0 $\le$ t $\le$ 2 $\pi$

Question 32

Question 33

Find g(t) so that x = -1 + 3 cos(t), y = g(t), 0 $\le$ t $\le$ 2 $\pi$ provides a counterclockwise parametrization of the circle + + 2x - 4y = 4.

A) -2 + sin(t)
B) 2 - 3 sin(t)
C) 2 + 3 sin(t)
D) 3 - 2 sin(t)
E) 3 + 2 sin(t)

Question 34

Which of the following best describes the parametric curve x = sec(t), y = (t),- $\le$ t $\le$ ?

A) part of the parabola y = x² - 1 lying under the line y = 1
B) the parabola y = x² - 1
C) part of the parabola y = x² - 1 lying above the line y = 1
D) part of the parabola y = x² - 1 lying above the line y = -1
E) part of the parabola y = x² - 1 lying under the line y = 1 in the first quadrant

Question 35

Find the slope of the curve x = 6t + 3, y = 2 - 7t when t = 5.

A) 6
B) 3
C)
D)
E) 0

Question 36

Find the equation of the tangent line to the curve at the given t.
X = cos 3t, y = 3 sin 5t at t = .

A) x + y = 2
B) y = -
C) x = -1
D) x + y + 1 + = 0
E) x = 1

Question 37

Find the equation of the tangent line to the curve at the given t. x = , y = at t = 1.

A) x + y = 3
B) 2x + y = 5
C) 2x - y = 3
D) x - y = 1
E) x + 2y = 3

Question 38

Find the slope of the curve x = 5 cos t, y = 3 sin t at t = .

A)
B) -
C) -
D)
E) 1

Question 39

Find the equation of the tangent line to the curve at the given t. x = 2 cot t, y = 2 t at t =

A) x - 2y = 0
B) x + 2y = 4
C) 2x + y = 5
D) 2x - y = 3
E) x - 2y = 4

Question 40

Find the Cartesian equation of the straight line tangent to the plane curve given parametrically by at the point on the curve where t = -1.

A) x + y = 0
B) 3x - y =0
C) y = 0
D) y = x
E) y = -3x

Question 41

Express and in terms of x and y for the circle x = a cos $\theta$
, y = a sin $\theta$
.

A) = - , = -
B) = , =
C) = - , = -
D) = , = -
E) = , = -

Question 42

Determine the points where the parametric curve x = - 3t, y = - 12t have horizontal and vertical tangents.

A) horizontal tangents at (2, 11), (-2, -11) and vertical tangents at (2, -16), (-2, 16)
B) horizontal tangent at (0, 0)
C) vertical tangent at (0, 0)
D) horizontal tangents at (2, -16), (-2, 16) and vertical tangents at (2, 11), (-2, -11)
E) no horizontal or vertical tangents

Question 43

Find equations of the three normal lines to the parabola given parametrically by the equations
x(t) = , y(t) = 2t, which pass through the point P (3, 0).

Question 44

Find at the highest point on the cycloid x = a $\theta$
- a sin $\theta$
, y = a - a cos $\theta$
.

A) -
B) -
C)
D)
E)

Question 45

Find the slope of the curve x = sin 2t, y = cos 3t at t = 0.

A) -
B) -
C) 0
D)
E)

Question 46

Find the coordinates of the highest point of the curve x = 6t, y = 6t - .

A) (18, 9)
B) (0, 0)
C) (12, 6)
D) (6, 5)
E) (24, 8)

Question 47

Find the slopes of two lines tangent to the parametric curve x = + - 6t + 1,y = + t - 4 at the point on the curve.

A) 2, -3
B) - ,
C) -6, -
D) , -
E) - ,

Question 48

Where does the curve x = 2 - 5, y = + t have a tangent line that is perpendicular to the line ?

A) and (-3, 2)
B) and (3, -2)
C) and (3, -2)
D) and (-3, 2)
E) and (-3, -2)

Question 49

Where is the curve x = ln t, y = e^t concave upward?

A) at all points on the curve
B) at all points corresponding to values of t satisfying t > 1
C) at all points corresponding to values of t satisfying 0 < t < 1
D) at all points corresponding to values of t satisfying 0 < t $\le$ 1
E) nowhere

Question 50

Find the slope of the curve x = 3 csc(t), y = 2 cot(t) at the point t = .

A)
B) -
C) -
D)
E)

Question 51

Find the Cartesian equation of the straight line tangent to the plane curve given parametrically by the equations x(t) = + 2t + 2, y(t) = 1 - 3 - 2 at the point on the curve where t = -1.

A) x -3y -1 = 0
B) y =1
C) y = x + 1
D) 3x -y -3 = 0
E) y = 3x -7

Question 52

At what values of t does the curve x = t - sin t, y = 1 - cos t have (a) a horizontal tangent, (b) a vertical tangent, and (c) no tangent?

A) (a) t = 2k $\pi$ , k is an integer; (b) t = (2k + 1) $\pi$ ; (c) nowhere
B) (a) t = k $\pi$ , k is an integer; (b) nowhere; (c) nowhere
C) (a) t = (2k + 1) $\pi$ , k is an integer; (b) t = 2k $\pi$ , k is an integer; (c) nowhere
D) (a) t = (2k + 1) $\pi$ , k is an integer; (b) nowhere; (c) t = 2k $\pi$ , k is an integer
E) (a) t = k $\pi$ , k is an integer; (b) t = 2k $\pi$ , k is an integer; (c) nowhere

Question 53

Find the tangent line(s) to the parametric curve given by x = - 4 , y= at (0, 4).

A) y = 4 ± x
B) y = 4 ± x
C) y = 8 ± x
D) y = 8 ± x
E) y = 2 ± x

Question 54

Determine the coordinates of the points where the curve x = + 2t, y = 2 + 7 has (a) a horizontal tangent and (b) a vertical tangent.

Question 55

Determine the coordinates of the points where the curve x = + 2t, y = 2 + 7 has (a) a horizontal tangent and (b) a vertical tangent.

A) (a) (0, -9) (b) (±2, -6)
B) (a) (-2, -9) (b) (-2, -6)
C) (a) (2, -9) (b) (2, -6)
D) (a) (0, -9) (b) (2, 6)
E) (a) (2, 9) (b) (2, 6)

Question 56

Find the arc length of x = u, y = , 0 $\le$ u $\le$ .

A) units
B) 2 units
C) units
D) units
E) units

Question 57

Find the length of the curve x = cos t + sin t, y = sin t - cos t, from t = to t = .

A) units
B) units
C) units
D) units
E) units

Question 58

Find the length of x = ln sin $\theta$ , y = $\theta$ , $\le$ $\theta$ $\le$ .

A) ln (4 - ) units
B) ln (3 + ) units
C) ln (2 + ) units
D) ln (1 + 2 ) units
E) ln (1 - 3 ) units

Question 59

A) 10 units
B) 15 units
C) 20 units
D) 25 units
E) 16 units

Question 60

Find the arc length of the curve x = e^t sin t, y = e^t cos t, from t = - to t = .

A) - 1) units
B) 2 + 1) units
C) 2 - 1) units
D) 2 sinh units
E) 2 cosh units

Question 61

Find the arc length of the curve x = ln (1 + ), y = t, from t = 0 to t = 1.

A) ln (2 - ) units
B) ln (3 - ) units
C) ln ( + 1) units
D) ln ( - 1) units
E) ln ( ) units

Question 62

Find the length of one arch of the cycloid x = a( $\theta$ - sin $\theta$ ), y = a(1 - cos $\theta$ ).

A) 8a units
B) 10a units
C) 12a units
D) 6a units
E) 4a units

Question 63

Find the area of the surface generated by rotating x = t - sin t, y = 1 - cos t where t [0, 2 $\pi$ ] about the x-axis.

A) square units
B) square units
C) square units
D) square units
E) 64 $\pi$ square units

Question 64

Find the area of the surface generated by rotating the astroid x = a t, y = a t about .

A) square units
B) square units
C) square units
D) square units
E) square units

Question 65

Find the area of the surface generated by rotating x = , y = , 0 $\le$ t $\le$ 1 about the y-axis.

A) 4 $\pi$ (5 - 8) square units
B) 4 $\pi$ (5 + 8) square units
C) 2 $\pi$ (5 - 8) square units
D) 2 $\pi$ (5 + 8) square units
E) $\pi$ (5 - 8) square units

Question 66

Find the length of the curve x = t, y = , 0 $\le$ t $\le$ 1.

A) 1 + ln( + 1) units
B) + ln( + 1) units
C) - ln( + 1) units
D) 1 - ln( + 1) units
E) 1 + ln( ) units

Question 67

Find the area of the surface generated by rotating x = t, y = , 0 $\le$ t $\le$ 1, about the x-axis.

A) square units
B) square units
C) square units
D) square units
E) square units

Question 68

Find the arc length of the curve x = , y = dx, 0 ≤ t ≤ ln(2).

Question 69

Find the arc length x = 2 cos $\theta$ + cos 2 $\theta$ + 1, y = 2 sin $\theta$ + sin 2 $\theta$ , for 0 $\le$ $\theta$$\le$ 2 $\pi$ .

A) 12 units
B) 14 units
C) 16 units
D) 18 units
E) 10 units

Question 70

Find the area of the region bounded by the ellipse x = 7 cos $\theta$
, y = 9 sin $\theta$ .

A) 63 $\pi$ square units
B) 16 $\pi$ square units
C) 2 $\pi$ square units
D) 25 $\pi$ square units
E) 72 $\pi$ square units

Question 71

Find the area of the region bounded by the hypocycloid x = a $\theta$
, y = a $\theta$
.

A) $\pi$ square units
B) $\pi$ square units
C) $\pi$ square units
D) $\pi$ square units
E) $\pi$ square units

Question 72

Determine the area above the x-axis and under one arch of the cycloid x = a(t - sin t), y = a(1 - cos t).

A) 2 $\pi$ square units
B) 3 $\pi$ square units
C) 4 $\pi$ square units
D) 6 $\pi$ square units
E) 5 $\pi$ square units

Question 73

Find the rectangular coordinates of the point with polar coordinates .

A) ( , 1)
B) (1, )
C) ( , 2)
D) (2, )
E) (3, 1)

Question 74

Convert the point with Cartesian coordinates (-1, -1) to polar coordinates.

A)
B)
C)
D)
E)

Question 75

Find a rectangular equation equivalent to the polar equation $\theta$
= .

A) y = x
B) y = x
C) y = x
D) y = 2x
E) y = 3x

Question 76

Find a rectangular equation equivalent to the polar equation r = tan $\theta$
.

A) =
B) =
C) =
D) =
E) =

Question 77

The equation of a conic section in polar coordinates is given by r = .(i) Transform the equation of the conic section to rectangular coordinates (x , y).(ii) Identify the conic section.

Question 78

Convert x² + y² = xy to polar coordinates.

A) r² =
B) r =
C) r =
D) r² =
E) r² =

Question 79

Describe the plane curve represented in polar coordinates (r , $\theta$ ) by the equation r = 4 sin( $\theta$ ) , $\theta$ [0 , $\theta$ ].

A) a straight line through the origin of slope 4
B) a circle centred at (x , y) = (2 , 0) and is of radius 2
C) a circle centred at (x , y) = (0 , 2) and is of radius 2
D) a circle centred at (x , y) = (0 , 2) and is of radius 4
E) a circle centred at (x , y) = (2 , 0) and is of radius 4

Question 80

Describe the graph of the polar equation r = 6(sin $\theta$ + cos $\theta$ ).

A) Straight line with intercepts (6, 0) and (0, 6)
B) Circle with centre at (3, 3) and radius 6
C) Circle with centre at (3, 3) and radius 3
D) Straight line with intercepts (3, 0) and (0, 3)
E) Circle with centre at (3, 3) and radius