Question 1

Find the equation of the circle that satisfies the following conditions. Express the final equation in the form x ² + y ² + Dx + Ey + F = 0. Center at (0, -1) and r = 6 A) x ² + y ² - 2 y - 37 = 0 B) x ² + y ² - 2 y + 1 = 0 C) x ² + y ² + 2 y - 35 = 0 D) x ² + y ² + 2 y + 1 = 0

Accepted Answer

The general equation of a circle with center at (h, k) and radius r is given by $(x-h)^2 + (y-k)^2 = r^2$. For a circle centered at (0, -1) with radius 6, the equation becomes $x^2 + (y+1)^2 = 36$, which simplifies to $x^2 + y^2 + 2y + 1 - 36 = 0$, or $x^2 + y^2 + 2y - 35 = 0$.

Question 2

Select the graph of the ellipse. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

A

Question 3

One section of a suspension bridge hangs between two towers that are 40 feet above the surface and 300 feet apart. A cable strung between the tops of the two towers is in the shape of a parabola with its vertex 20 feet above the surface. With axes drawn as indicated in the figure, find the equation of the parabola. A) x ² - 563 y + 45,000 = 0 B) x ² - 1,125 y + 22,500 = 0 C) x ² - 695 y + 55,600 = 0 D) x ² - 1,390 y + 27,800 = 0 E) x ² - 1,100 y + 22,000 = 0

Accepted Answer

B

Question 4

A parabolic arch spans a stream 220 feet wide. How high above the stream must the arch be to give a minimum clearance of 20 feet over a channel in the center that is 180 feet wide?
A) 20 feet
B) 31.325 feet
C) 60.5 feet
D) 80.5 feet
E) 34.725 feet

Accepted Answer

The arch must be high enough so that the lowest point of the arch over the 180-foot wide channel is at least 20 feet. The parabola's vertex is at the center of the 220-foot span, so the equation is y = a(x - 110)^2 + h. The channel extends 90 feet on either side of the center, so at x = 20 and x = 200, y must be 20 feet. Solving for h gives h = 60.5 feet.

Question 5

Find the center and the length of a radius of the circle. x ² + y ² - 8 x + 2 y = 0 A) B) C) D) E)

Accepted Answer

The answer of Find the center and the length of...

Question 6

Write the equation of the circle that satisfies the following conditions. Express the final equation in the form x ² + y ² + Dx + Ey + F = 0. Center at the origin and r = 4 A) x ² + y ² = 0 B) x ² - y ² - 16 = 0 C) x ² - y ² + 16 = 0 D) x ² + y ² + 16 = 0 E) x ² + y ² - 16 = 0

Accepted Answer

The equation of a circle with center at the origin and radius $r$ is $x^2 + y^2 = r^2$. With $r = 4$, $r^2 = 16$, so the equation is $x^2 + y^2 - 16 = 0$.

Question 7

Refer to the illustration. How far does a point on the tip of a rotor blade travel when it makes one complete revolution? The length of the blade  is equal to 17 feet. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Refer to the illustration. How far does...

Question 8

Find the center and radius of a circle given an equation in standard form. 
A) 
B) 
C) 
D)

Accepted Answer

The answer of Find the center and radius of a...

Question 9

If the orbit of the comet is approximated by the equation  , how far is it from the sun at the vertex of the orbit? Distances are measured in astronomical units (AU).  __________ AU

Accepted Answer

The answer of If the orbit of the comet is...

Question 10

Find the equation of the circle that satisfies the following conditions. Express the final equation in the form x ² + y ² + Dx + Ey + F = 0. Tangent to the x axis, a radius of length 2, and abscissa of center is -1 A) x ² + y ² - 2 x - 4 y + 1 = 0 and x ² + y ² - 2 x + 4 y + 1 = 0 B) x ² + y ² + 2 x - 4 y + 1 = 0 C) x ² + y ² + 2 x - 4 y + 1 = 0 and x ² + y ² + 2 x + 4 y + 1 = 0 D) x ² + y ² + 2 x + 4 y + 1 = 0 E) x ² + y ² - 2 x + 4 y + 1 = 0

Accepted Answer

Since the circle is tangent to the x-axis and has a radius of length 2, we know that the center of the circle lies on the line x=-1 and is 2 units away from the x-axis. This means that the y-coordinate of the center is -2. 
So the center of the circle is (-1,-2) and the equation of the circle can be written in the form:
(x+1)^2 + (y+2)^2 = 4 
Expanding this equation and rearranging, we get:
x^2 + y^2 + 2x + 4y + 1 = 0

So, the correct answer is option C, which includes this equation as one of its choices.

Question 11

Joan wants to jog  miles on a circular track  mile in diameter. How many times must she circle the track?
A) 45.84
B) 11.46
C) 22.92
D) 5.73
E) 2.86

Accepted Answer

The circumference of a circle is given by the formula C = πd, where d is the diameter. In this case, the diameter is 1 mile, so the circumference is C = π(1) = π miles. To jog 11 miles, Joan must circle the track 11/π = 3.5 times. However, since Joan cannot jog a fraction of a lap, she must round up to the nearest whole number of laps, which is 4. Therefore, Joan must circle the track 4 times, which is equivalent to jogging 4 times the circumference of the track, or 4π miles.

To confirm that choice C is the correct answer, we can also calculate the distances for the other choices: 
A) 45.84π miles is far too long, as it is nearly 144 miles. 
B) 11.46π miles is also too long, as it is nearly 36 miles. 
D) 5.73π miles is too short, as it is less than 18 miles. 
E) 2.86π miles is also too short, as it is less than 9 miles. Therefore, the only correct answer is C.

Question 12

Find the length of a radius of this circle. x ² + y ² = 13 A) B) C) D) E)

Accepted Answer

The answer of Find the length of a radius of...

Question 13

Graph the equation. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Graph the equation. 
A) 
B) 
C) 
D)...

Question 14

Find the center and radius of a circle given an equation in standard form. 
A) C (0, 0), 
B) C (0, 0), 
C) C (0, 0), 
D) C (0, 5),

Accepted Answer

The equation is already in standard form, which is of the form $(x-a)^2+(y-b)^2=r^2$, where $(a,b)$ is the center and $r$ is the radius.
Comparing this to the given equation, we have:
$(x-0)^2 + (y-0)^2 = 11ec1e8e_2bcd_a77d_be9b_41a328ca6e31_TBX8967_11$
So the center is at $(0,0)$ and the radius is $\sqrt{11ec1e8e_2bcd_a77d_be9b_41a328ca6e31_TBX8967_11}$. 
Since none of the other answer choices match this, the answer is B.

Question 15

Find the center and radius of a circle given an equation in standard form. 
A) 
B) 
C) 
D)

Accepted Answer

The equation of a circle in standard form is $(x-a)^2+(y-b)^2=r^2$, where $(a,b)$ is the center and $r$ is the radius. 
By comparing the given equation to this standard form, we can see that $(x+2)^2+(y-1)^2=4^2$. This means the center is $(-2,1)$ and the radius is $4$. Therefore, the correct choice is $\boxed{	extbf{(C)}}$.

Question 16

Find the vertex of the parabola and graph it. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find the vertex of the parabola and...

Question 17

Find the vertex of the parabola and graph it. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find the vertex of the parabola and...

Question 18

Suppose the two "legs" of the compass shown in the illustration are adjusted so that the distance between the pointed ends is 3 inches. What will the area of the circle be, if  ? 
A) 28.26
B) 18.84
C) 3.1415926536
D) 6
E) none of these

Accepted Answer

First, we can use the formula for the area of a circle:

A = πr^2

Next, we need to find the radius of the circle. We can do this by finding the midpoint of the line segment between the pointed ends of the compass, which will be the center of the circle.

Since the distance between the pointed ends of the compass is 3 inches, the midpoint will be 1.5 inches away from each end.

We can use the Pythagorean theorem to find the distance from the midpoint to one of the pointed ends:

a^2 + b^2 = c^2

(1.5)^2 + (3/2)^2 = r^2

r ≈ 1.837 inches

Now we can plug this value into the formula for the area of a circle:

A = π(1.837)^2

A ≈ 28.26

Therefore, the answer is A) 28.26.

Question 19

Find the center and the length of a radius of the circle. x ² + y ² - 2 x - 6 y + 1 = 0 A) (-1, -3), r = 3 B) (3, -1), r = 6 C) (1, 3), r = 6 D) (3, 1), r = 3 E) (1, 3), r = 3

Accepted Answer

The equation can be rewritten in the standard form of a circle by completing the square: (x-1)^2 + (y-3)^2 = 3^2. Thus, the center is (1, 3) and the radius is 3.

Question 20

Find the vertex of the parabola and graph it. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find the vertex of the parabola and...

Quiz 10: Conic Sections and More Graphing

Find the equation of the circle that satisfies the following conditions. Express the final equation in the form x 2 + y 2 + Dx + Ey + F = 0. Center at (0, -1) and r = 6

Select the graph of the ellipse.

A parabolic arch spans a stream 220 feet wide. How high above the stream must the arch be to give a minimum clearance of 20 feet over a channel in the center that is 180 feet wide?

Find the center and the length of a radius of the circle. x 2 + y 2 - 8 x + 2 y = 0

Write the equation of the circle that satisfies the following conditions. Express the final equation in the form x 2 + y 2 + Dx + Ey + F = 0. Center at the origin and r = 4

Refer to the illustration. How far does a point on the tip of a rotor blade travel when it makes one complete revolution? The length of the blade is equal to 17 feet.

Find the center and radius of a circle given an equation in standard form.

If the orbit of the comet is approximated by the equation , how far is it from the sun at the vertex of the orbit? Distances are measured in astronomical units (AU). __________ AU

Find the equation of the circle that satisfies the following conditions. Express the final equation in the form x 2 + y 2 + Dx + Ey + F = 0. Tangent to the x axis, a radius of length 2, and abscissa of center is -1

Joan wants to jog miles on a circular track mile in diameter. How many times must she circle the track?

Find the length of a radius of this circle. x 2 + y 2 = 13

Graph the equation.

Find the center and radius of a circle given an equation in standard form.

Find the center and radius of a circle given an equation in standard form.

Find the vertex of the parabola and graph it.

Find the vertex of the parabola and graph it.

Suppose the two "legs" of the compass shown in the illustration are adjusted so that the distance between the pointed ends is 3 inches. What will the area of the circle be, if ?

Find the center and the length of a radius of the circle. x 2 + y 2 - 2 x - 6 y + 1 = 0

Find the vertex of the parabola and graph it.

Find the equation of the circle that satisfies the following conditions. Express the final equation in the form x ² + y ² + Dx + Ey + F = 0. Center at (0, -1) and r = 6

Find the center and the length of a radius of the circle. x ² + y ² - 8 x + 2 y = 0

Write the equation of the circle that satisfies the following conditions. Express the final equation in the form x ² + y ² + Dx + Ey + F = 0. Center at the origin and r = 4

Find the equation of the circle that satisfies the following conditions. Express the final equation in the form x ² + y ² + Dx + Ey + F = 0. Tangent to the x axis, a radius of length 2, and abscissa of center is -1

Find the length of a radius of this circle. x ² + y ² = 13

Find the center and the length of a radius of the circle. x ² + y ² - 2 x - 6 y + 1 = 0