Deck 14: Parametric Equations and Conic Sections

Question 1

Describe the graph of the parametric equations $x=2 \cos \left(\frac{\pi}{t+1}\right), y=2 \sin \left(\frac{\pi}{t+1}\right)$ for $t>$ 0.

A) A circle of radius 2, centered at the origin and traced in the clockwise direction.
B) The right half of a circle of radius 2, centered at the origin, traced in the clockwise direction, and never quite reaching the point $(0,2)$ .
C) The lower half of a circle of radius 2 , centered at the origin, traced in the counterclockwise direction, and never quite reaching the point $(2,0)$ .
D) The upper half of a circle of radius 2, centered at the origin, traced in the clockwise direction, and never quite reaching the point $(2,0)$ .

Answer 1

The upper half of a circle of radius 2, centered at the origin, traced in the clockwise direction, and never quite reaching the point $(2,0)$ .

Question 2

Let $y=t$ be one part of a parametric equation. Which of the following formulas for $x$ as a function of $t$ on $0 \leq t \leq 6$ completes the parameterization of the graph of a straight line from $(0,0)$ to $(1,1)$ , and then continuing from $(1,1)$ in a straight line to $(0,6)$ ?

A) $x=\left\{\begin{array}{cc}t & 0 \leq t \leq 1 \\ 6-t & 1 \leq t \leq 6\end{array}\right.$
B) $x=\left\{\begin{array}{cc}6-t & 0 \leq t \leq 1 \\ t & 1 \leq t \leq 6\end{array}\right.$
C) $x= \begin{cases}t & 0 \leq t \leq 1 \\ 6 & 1 \leq t \leq 6\end{cases}$
D) $x=\left\{\begin{array}{cc}1-t & 0 \leq t \leq 1 \\ 6 & 1 \leq t \leq 6\end{array}\right.$

Answer 2

$x=\left\{\begin{array}{cc}t & 0 \leq t \leq 1 \\ 6-t & 1 \leq t \leq 6\end{array}\right.$

Question 3

Could the graph below be the graph of the parametric equations $x=-t+3, y=\sqrt{t}$ ? Assume the parameter is restricted to values for which the functions are defined.

Answer 3

False

Question 4

Could the graph below be the graph of the parametric equations $x=e^{t}, y=t$ ? Assume the parameter is restricted to values for which the functions are defined.

Question 5

Could the graph below be the graph of the parametric equations $x=\cos t, y=2.2 \sin t$ ? Assume the parameter is restricted to values for which the functions are defined.

Question 6

What is the explicit formula for the curve $x=t^{5}+3, y=t^{4}+5$ ?

A) $y=(x-5)^{4 / 5}+3$
B) $y=(x-3)^{4 / 5}+5$
C) $y=(x+5)^{5 / 4}+3$
D) $y=(x-3)^{5 / 4}-5$

Question 7

Describe the graph of the curve $x=t^{4}+6, y=t^{3}+7$ .

A) The graph looks like the graph of $y=x^{4 / 3}$ translated 6 units to the right and 7 units up.
B) The graph looks like the graph of $y=x^{4 / 3}$ translated 7 units to the right and 6 units up
C) The graph looks like the graph of $y=x^{3 / 4}$ translated 6 units to the right and 7 units up.
D) The graph looks like the graph of $y=x^{3 / 4}$ translated 7 units to the right and 6 units up.

Question 8

What is the explicit formula for the curve $x=e^{0.1 t}, y=e^{t}, 0 \leq t \leq 1$ ?

A) $y=e^{10 x}$
B) $y=e^{0.1 x}$
C) $y=x e^{0.1}$
D) $y=x^{10}$

Question 9

Describe the graph of the parametric equations $x=e^{6 t}, y=e^{12 t},-\infty<t<\infty$ .

A) The parabola $y=x^{2}$ .
B) The right half of the parabola $y=x^{2}$ .
C) The circle $x^{2}+y^{2}=36$ .
D) The top half of the circle $x^{2}+y^{2}=36$ .

Question 10

Is the equation $y=x^{2}+4$ explicit or implicit?

Question 11

What are the endpoints for the curve $x=e^{0.05 t}, y=e^{t}, 0 \leq t \leq 1$ ? Mark both correct answers.

A) $(1,1)$
B) $(\sqrt[20]{e}, e)$
C) $(e, e)$
D) $(0,0)$

Question 12

Give a parameterization for the straight line segment joining the points $(6,4)$ and $(-3,-2)$ .

Question 13

Give a parameterization for the upper half circle of radius 3 centered at $(7,-2)$ , and starting at $(10,-2)$

Question 14

Give a parameterization for the vertical line through the point $(2,-4)$ .

Question 15

Give a parameterization for the horizontal line through the point $(3,4)$ .

Question 16

The line between $(0,6)$ and $(-4,11)$ can be parameterized by $x=-4 t, y=6+5 t$ , $0 \leq t \leq 1$ .

Question 17

The line between $(0,4)$ and $(-4,11)$ can be parametrized by $x=-4 t, y=4+11 t$ , $0 \leq t \leq 1$ .

Question 18

As $t$ varies, the equations $x=2+6 \cos t$ and $y=-3+6 \sin t$ trace out a circle.
a) Describe the center and radius of the circle.
b) If $0 \leq t \leq \frac{\pi}{2}$ , what part of the circle is obtained?

Question 19

Let $x=\sqrt{3 t}, y=8 t+8,0 \leq t \leq \infty$ .
a) Graph the curve.
b) Eliminate the parameter to obtain an equation for $y$ as a function of $x$ .

Question 20

Let $x=e^{2 t}, y=e^{5 t},-\infty<t<\infty$ .
a) Graph the curve.
b) Eliminate the parameter to obtain an equation for $y$ as a function of $x$ .

Question 21

Choose the parametric equations that describe the curve

A) $x=\cos 6 t$ and $y=\cos 4 t$
B) $x=\sin 6 t$ and $y=\cos 4 t$
C) $x=\cos 6 t$ and $y=\sin 4 t$
D) $x=\sin 6 t$ and $y=\sin 4 t$

Question 22

Choose the parametric equations that describe the curve

A) $x=4 \cos t-\cos 8.8 t$ and $y=4 \sin t-\sin 6.5 t$
B) $x=4 \cos t+\sin 8.8 t$ and $y=4 \sin t+\cos 6.5 t$
C) $x=4 \cos t+\cos 8.8 t$ and $y=4 \sin t+\sin 6.5 t$
D) $x=4 \sin t+\sin 6.5 t$ and $y=4 \cos t+\cos 8.8 t$

Question 23

A mouse hanging on to the end of the windmill blade shown below has coordinates given by $x=A \cos (k t), y=A \sin (k t)$ , where the origin is at the center of the blades, $x$ and $y$ are in meters, and $t$ is in seconds. The blades are 6 meters long and the windmill makes one complete revolution every 20 seconds in a counterclockwise direction. The mouse starts in the 3 o'clock position and, 7.5 seconds later, loses its hold and flies off. What is $A$ ?

Question 24

A mouse hanging on to the end of the windmill blade shown below has coordinates given by $x=A \cos (k t), y=A \sin (k t)$ , where the origin is at the center of the blades, $x$ and $y$ are in meters, and $t$ is in seconds. The blades are 7 meters long and the windmill makes one complete revolution every 32 seconds in a counterclockwise direction. The mouse starts in the 3 o'clock position and, 4 seconds later, loses its hold and flies off. What is $k$ ? Round to 2 decimal places.

Question 25

A mouse hanging on to the end of the windmill blade shown below has coordinates given by $x=A \cos (k t), y=A \sin (k t)$ , where the origin is at the center of the blades, $x$ and $y$ are in meters, and $t$ is in seconds. The blades are 3 meters long and the windmill makes one complete revolution every 20 seconds in a counterclockwise direction. The mouse starts in the 3 o'clock position and, 7.5 seconds later, loses its hold and flies off. How many meters\sec. is the mouse traveling while it is on the blade? Round to 2 decimal places.

Question 26

A mouse hanging on to the end of the windmill blade shown below has coordinates given by $x=A \cos (k t), y=A \sin (k t)$ , where the origin is at the center of the blades, $x$ and $y$ are in meters, and $t$ is in seconds. The blades are 5 meters long and the windmill makes one complete revolution every 28 seconds in a counterclockwise direction. The mouse starts in the 3 o'clock position and, 3.5 seconds later, loses its hold and flies off. How many degrees is the angle between the blade and the positive $x$ -axis at the moment the mouse flies off?

Question 27

A mouse hanging on to the end of the windmill blade shown below has coordinates given by $x=A \cos (k t), y=A \sin (k t)$ , where the origin is at the center of the blades, $x$ and $y$ are in meters, and $t$ is in seconds. The blades are 7 meters long and the windmill makes one complete revolution every 28 seconds in a counterclockwise direction. The mouse starts in the 3 o'clock position and, 10.5 seconds later, loses its hold and flies off. Assume that when the mouse leaves the blade it moves along a straight line tangent to the circle on which it was previously moving. The equation of that line is y=--------+-----------x. Round to 2 decimal places if necessary.

Question 28

If the circle $x^{2}+y^{2}+12 x-16 y+51=0$ is put into standard form
$(x-h)^{2}+(y-k)^{2}=r^{2}$ , then $h=\ldots, k=\ldots$ , and $r=$ ------.

Question 29

If the circle $x^{2}+y^{2}+14 x-6 y+9=0$ is converted to the parametric equations $x=h+r \cos t, y=k+r \sin t, 0 \leq t \leq 2 \pi$ , then h=------------,k=----------,and r=------------.

Question 30

The center of the circle $5(x-7)^{2}+4(y+4)^{2}-125=-(y+4)^{2}$ is at $(\ldots, \ldots)$ .

Question 31

What is the radius of the circle $5(x-5)^{2}+4(y+6)^{2}-45=-(y+6)^{2}$ ?

Question 32

Let $x=a+k \cos t$ and $y=b+k \sin t$ parameterize a circle, with $a<0, b<0, k<a$ , and $k<b$ . What quadrant does the circle lie in?

A) The second
B) The third
C) The fourth
D) The first

Question 33

What is the radius of the circle parameterized by $x=9 \cos t, y=9 \sin t, 0 \leq t \leq 2 \pi$ ?

Question 34

What is the direction of the circle parameterized by $x=4 \cos t, y=4 \sin t, 0 \leq t \leq 2 \pi$ ?

A) counterclockwise
B) clockwise

Question 35

The center of the circle parameterized by $x=4+9 \cos t, y=-1+9 \sin t, 0 \leq t \leq 2 \pi$ is at (----------,---------).

Question 36

Which of the following sets of equations parameterize the circle $x^{2}+y^{2}=25$ ? Mark all that apply.

A) $x=5 \cos t, y=5 \sin t, 0 \leq t \leq 2 \pi$
B) $x=-5 \cos t, y=-5 \sin t, 0 \leq t \leq 2 \pi$
C) $x=5 \cos ^{2} t, y=5 \sin ^{2} t, 0 \leq t \leq 2 \pi$
D) $x=\cos 5 t, y=\sin 5 t, 0 \leq t \leq 2 \pi$
E) $x=5 \cos 2 t, y=5 \sin 2 t, 0 \leq t \leq \pi$

Question 37

Parameterize the circle of diameter 4 centered at $(6,-3)$ traversed counterclockwise starting at $(8,-3)$ .

Question 38

A bug starts at the point $(1,-1)$ and moves at 5 units\second along the $y$ -axis to the point $(1,2)$ . Then, the ant moves clockwise along a circle of radius 1 centered at $(1,1)$ to the point $(1,0)$ at a speed of 4 units\second. Express the bug's coordinates as a function of time, $t$ , in seconds.

Question 39

Identify the center and radius of
$4 x^{2}-8 x+4 y^{2}-24 y=-35$

Question 40

$(x-4)^{2}+(y+3)^{2}=16$ is a circle of radius 4 centered at $(-4,3)$

Question 41

$(x-3)^{2}+(y+5)^{2}=9$ is a circle of radius 3 centered at $(3,-5)$ .

Question 42

What curve is parameterized by the functions $x=4+\sin t$ and $y=4+\cos t$ ? Give an implicit or explicit equation for the curve.

Question 43

Is the function $4 x^{2}+3 x^{3}=2$ implicit or explicit?

Question 44

Is the function $4 x^{3}+4=y$ implicit or explicit?

Question 45

Select the type of curve formed by the parametric equations $x=2+6 \cos 2 t$ and $y=-1+3 \sin 2 t$ .

A) Circle
B) Parabola
C) Elipse
D) Line

Question 46

Select the type of curve formed by the parametric equations $x=2+3 \cos t$ and $y=2+3 \sin t$ .

A) Circle
B) Parabola
C) Elipse
D) Line

Question 47

Select the type of curve formed by the parametric equations $x=2-\frac{4}{\cos 5 t}$ and $y=2-6 \sin 5 t$ .

A) Circle
B) Parabola
C) Elipse
D) Hyperbola

Question 48

Select the type of curve formed by the parametric equations $x=4+\cos ^{2} t$ and $y=-3-\sin t$ .

A) Circle
B) Parabola
C) Elipse
D) Hyperbola

Question 49

If the ellipse $x^{2}+9 y^{2}-4 x-72 y+139=0$ is written in standard form $\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1$ , then h=-----------,k=----------,a=-----------,and b=---------.

Question 50

Are the parametric equations for the quarter of an ellipse centered at $(0,0)$ , starting at $(0,2)$ and ending at $(-4,0)$ , given by $x=4 \cos t, y=2 \sin t, \frac{\pi}{2} \leq t \leq \pi$ ?

Question 51

Find the upper intersection point of the ellipse $\frac{(x+3)^{2}}{9}+\frac{(y-3)^{2}}{16}=1$ and the line $x=$ -2 . Round the $y$ -coordinate to 2 decimal points.

Question 52

Which of the following is the implicit equation for the ellipse graphed below?

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

Question 53

Does $x=-1+2 \sin t, y=1-3 \cos t, 0 \leq t \leq 2 \pi$ , parameterize the ellipse graphed below, traversed in the clockwise direction and starting at the point $(-1,-2)$ ?

Question 54

In the ellipse given by the equations $x=7+5 \cos t, y=4+3 \sin t, 0 \leq t \leq 2 \pi$ , the $x$ -values range from a minimum of ----------- to a maximum of ----------- about the midline x=----------.

Question 55

The ellipse given by the equations $x=2+3 \cos t, y=2+5 \sin t, 0 \leq t \leq 2 \pi$ , has a height of ---------- units and a width of ---------- units.

Question 56

The ellipse given by the equations $x=8+4 \cos t, y=4+5 \sin t, 0 \leq t \leq 2 \pi$ , has a minor axis of length ------------- units.

Question 57

In the ellipse given by the equation $\frac{(x-7)^{2}}{16}+\frac{(y-7)^{2}}{25}=1$ , the $x$ -values range from a minimum of --------- to a maximum of ------------about the midline x=------------.

Question 58

The ellipse given by the equation $\frac{(x-1)^{2}}{25}+\frac{(y-8)^{2}}{16}=1$ has a height of ---------- units and a width of ---------- units.

Question 59

The ellipse given by the equation $\frac{(x-7)^{2}}{36}+\frac{(y-1)^{2}}{25}=1$ has a minor axis of length -----------.

Question 60

Find the center of the ellipse and the lengths of the major and minor axes:
$25 x^{2}-10 x+16 y^{2}+96 y=231$

Question 61

Find a parameterization of the curve $\frac{(x-2)^{2}}{25}+\frac{(y-3)^{2}}{16}=1$ so that the curve is traversed in a counter-clockwise direction starting at $(7,3)$ .

Question 62

Let an ellipse be parameterized by the equations
$\begin{aligned}& x=4+2 \cos t \\& y=3+4 \sin t\end{aligned}$
a) What is the center?
b) What is the length of the major axis?
c) What is the length of the minor axis?
d) Find an implicit equation of the curve.

Question 63

Let an ellipse be parameterized by the equations
$\begin{aligned}& x=-3+3 \sin t \\& y=4+5 \cos t\end{aligned} .$
a) What is the center?
b) What is the length of the major axis?
c) What is the length of the minor axis?
d) Find an implicit equation of the curve.

Question 64

Write the ellipse $4 x^{2}+16 x+8 y^{2}-64 y=-139$ in the form $\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1$ .

Question 65

The polar equation $r=\frac{5}{1-\frac{1}{2} \cos \theta}$ is an ellipse. What is the center of this ellipse?

Question 66

Parameterize the ellipse $\frac{x^{2}}{4^{2}}+\frac{(y-2)^{2}}{4^{2}}=1$ so that the parameterization starts at the point $(4,2)$ and travels clockwise.

Question 67

Find the equation of the ellipse centered at $(1,-3)$ with horizontal axis of length 8 and vertical axis of length 12 .

Question 68

Parameterize the ellipse centered at the point $(-3,4)$ with horizontal axis of length 4 and vertical axis of length 10 .

Question 69

Select the equations that describe the graph

A) $x=4+2 \cos t$ and $y=4-4 \sin t$
B) $x=4+2 \sin t$ and $y=4+4 \sin t$
C) $x=4+2 \cos t$ and $y=4+4 \sin t$
D) $x=4+2 \sin t$ and $y=4+4 \cos t$

Question 70

If the equation $-4 x^{2}+4 y^{2}+8 x-32 y+44=0$ is written in standard form $\frac{(x-h)^{2}}{a^{2}}-\frac{(y-k)^{2}}{b^{2}}=1$ or $\frac{(y-k)^{2}}{b^{2}}-\frac{(x-h)^{2}}{a^{2}}=1$ , then h =-----------,k=----------,a=---------,and b =--------------.

Question 71

What is the center of the hyperbola $-4 x^{2}+9 y^{2}+8 x-54 y+41=0$ ?

Question 72

Does the hyperbola $9 x^{2}-4 y^{2}+54 x-8 y+41=0$ open left-right or up-down?

Question 73

Is $(4,-1)$ a vertex of the hyperbola shown below?

Question 74

Is $x=3$ an asymptote of the hyperbola shown below?

Question 75

In the hyperbola given by the equations $x=2+7 \sec t, y=6+7 \tan t, 0 \leq t \leq 2 \pi$ , the left vertex is at (-----------,--------------).

Question 76

In the hyperbola given by the equations $x=4+6 \sec t, y=7+3 \tan t, 0 \leq t \leq 2 \pi$ , the asymptote that slopes upward has equation y=-------------x+----------. Round both answers to 2 decimal places.

Question 77

In the hyperbola given by the equation $\frac{(x-6)^{2}}{49}-\frac{(y-4)^{2}}{25}=1$ , the left vertex is at (-----------,-------------)

Question 78

In the hyperbola given by the equation $\frac{(x-4)^{2}}{25}-\frac{(y-5)^{2}}{4}=1$ , the asymptote that slopes upward has equation $y=\ldots x+\ldots$ . Round both answers to 2 decimal places.

Question 79

Parameterize the hyperbola $\frac{x^{2}}{3^{2}}-\frac{y^{2}}{2^{2}}=1$ .

Question 80

Parameterize the hyperbola $\frac{y^{2}}{4^{2}}-\frac{x^{2}}{5^{2}}=1$