Question 1

Use the given information about a circle to find its equation.
-Center $( - 1 , - 5 )$ and radius $\sqrt { 3 }$
A) $( x + 1 ) ^ { 2 } + ( y + 5 ) ^ { 2 } = \sqrt { 3 }$
B) $( x + 1 ) ^ { 2 } + ( y + 5 ) ^ { 2 } = 3$
C) $( x - 1 ) ^ { 2 } + ( y - 5 ) ^ { 2 } = 3$
D) $( x - 1 ) ^ { 2 } + ( y - 5 ) ^ { 2 } = \sqrt { 3 }$

Accepted Answer

The equation of a circle with center $(h, k)$ and radius $r$ is $(x - h)^2 + (y - k)^2 = r^2$. Given the center $(-1, -5)$ and radius $\sqrt{3}$, the equation becomes $(x + 1)^2 + (y + 5)^2 = (\sqrt{3})^2$, which simplifies to $(x + 1)^2 + (y + 5)^2 = 3$.

Question 2

Use the given information about a circle to write an equation of the circle in standard form. Graph the
circle.
-Center: $( 4,2 )$; Radius: 4
A) $( x + 4 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 16$

B) $( x + 4 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 4$

C) $( x - 4 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 4$

D) $( x - 4 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 16$

Accepted Answer

The answer of Use the given information about a circle...

Question 3

Use the given information about a circle to find its equation.
-Center $( 6 , - 7 )$ and radius of 5
A) $( x + 6 ) ^ { 2 } + ( y - 7 ) ^ { 2 } = 5$
B) $( x + 6 ) ^ { 2 } + ( y - 7 ) ^ { 2 } = 25$
C) $( x - 6 ) ^ { 2 } + ( y + 7 ) ^ { 2 } = 5$
D) $( x - 6 ) ^ { 2 } + ( y + 7 ) ^ { 2 } = 25$

Accepted Answer

The equation of a circle with center $(h, k)$ and radius $r$ is $(x - h)^2 + (y - k)^2 = r^2$. Given the center $(6, -7)$ and radius $5$, the equation becomes $(x - 6)^2 + (y + 7)^2 = 5^2$, which simplifies to $(x - 6)^2 + (y + 7)^2 = 25$.

Question 4

Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
-$$\text { Is the point } ( - 3 , - 1 ) \text { on the circle defined by } ( x + 3 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 9$$
A) No
B) Yes

Accepted Answer

The answer of Choose the one alternative that best completes...

Question 5

Find an equation of the circle with the given characteristics.
-$( - 6,0 )$ and radius 3
A) $( x - 6 ) ^ { 2 } + y ^ { 2 } = 3$
B) $( x + 6 ) ^ { 2 } + y ^ { 2 } = 9$
C) $( x + 6 ) ^ { 2 } = 9$
D) $( x - 6 ) ^ { 2 } = 3$

Accepted Answer

The equation of a circle with center $(h, k)$ and radius $r$ is $(x - h)^2 + (y - k)^2 = r^2$. For a circle centered at $(-6, 0)$ with radius 3, the equation is $(x + 6)^2 + y^2 = 3^2$, which simplifies to $(x + 6)^2 + y^2 = 9$.

Question 6

Determine the center and radius of the circle.
-$$( x + 2 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = 45$$
A) Center: $( - 2,3 )$; Radius: $3 \sqrt { 5 }$
B) Center: $( - 2,3 )$; Radius: 45
C) Center: $( 2 , - 3 )$; Radius: $3 \sqrt { 5 }$
D) Center: $( 2 , - 3 )$; Radius: 45

Accepted Answer

The equation of a circle in the form $(x - h)^2 + (y - k)^2 = r^2$ has a center at $(h, k)$ and a radius of $r$. The given equation $(x + 2)^2 + (y - 3)^2 = 45$ matches this form with $h = -2$, $k = 3$, and $r^2 = 45$, which means $r = \sqrt{45} = 3\sqrt{5}$.

Question 7

Use the given information about a circle to write an equation of the circle in standard form.
-The endpoints of a diameter are $( 3,11 )$ and $( - 9 , - 5 )$.
A) $( x - 3 ) ^ { 2 } + ( y - 11 ) ^ { 2 } = 100$
B) $( x + 3 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = 100$
C) $( x - 3 ) ^ { 2 } + ( y + 3 ) ^ { 2 } = 20$
D) $( x + 3 ) ^ { 2 } + ( y + 11 ) ^ { 2 } = 20$

Accepted Answer

The center of the circle is the midpoint of the diameter, which is calculated as $\left( \frac{3 + (-9)}{2}, \frac{11 + (-5)}{2} \right) = (-3, 3)$. The radius can be found by the distance formula between one endpoint and the center, which gives $r = \sqrt{(3 + 3)^2 + (11 - 3)^2} = 10$. Thus, the equation is $(x + 3)^2 + (y - 3)^2 = 100$.

Question 8

Solve the problem.
-Write an equation that represents the set of points that are 5 units from $( - 5 , - 7 )$.
A) $( x + 5 ) ^ { 2 } + ( y + 7 ) ^ { 2 } = 25$
B) $( x + 5 ) ^ { 2 } + ( y + 7 ) ^ { 2 } = 5$
C) $| x + 5 | + | y + 7 | = 5$
D) $| x - 5 | + | y - 7 | = 5$

Accepted Answer

The equation represents a circle centered at $(-5, -7)$ with a radius of 5. The formula for a circle is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius. Substituting $(-5, -7)$ for $(h, k)$ and $5$ for $r$, we get $(x + 5)^2 + (y + 7)^2 = 25$.

Question 9

Determine the center and radius of the circle.
-$$( x - 4 ) ^ { 2 } + ( y - 8 ) ^ { 2 } = 9$$
A) Center: $( 4,8 )$; Radius: 9
B) Center: $( 4,8 )$; Radius: 3
C) Center: $( - 4 , - 8 )$; Radius: 3
D) Center: $( - 4 , - 8 )$; Radius: 9

Accepted Answer

The equation of a circle in the form $(x - h)^2 + (y - k)^2 = r^2$ has a center at $(h, k)$ and a radius of $r$. In this case, $h = 4$, $k = 8$, and $r^2 = 9$, so the radius $r$ is $\sqrt{9} = 3$.

Question 10

Find an equation of the circle with the given characteristics.
-Center $( 8 , - 7 )$ and radius 3
A) $( x - 8 ) ^ { 2 } + ( y + 7 ) ^ { 2 } = 9$
B) $( x + 8 ) ^ { 2 } + ( y - 7 ) ^ { 2 } = 3$
C) $( x - 8 ) ^ { 2 } + ( y + 7 ) ^ { 2 } = 3$
D) $( x + 8 ) ^ { 2 } + ( y - 7 ) ^ { 2 } = 9$

Accepted Answer

The equation of a circle with center $(h, k)$ and radius $r$ is $(x - h)^2 + (y - k)^2 = r^2$. For a center at $(8, -7)$ and radius 3, the equation is $(x - 8)^2 + (y + 7)^2 = 3^2$, which simplifies to $(x - 8)^2 + (y + 7)^2 = 9$.

Question 11

Find an equation of the circle with the given characteristics.
-Center $( - 10 , - 2 )$ and radius 6
A) $( x - 10 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 6$
B) $( x + 10 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 6$
C) $( x + 10 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 36$
D) $( x - 10 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 36$

Accepted Answer

The general equation of a circle is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius. For a center at $(-10, -2)$ and radius $6$, the equation becomes $(x + 10)^2 + (y + 2)^2 = 36$.

Question 12

Use the given information about a circle to write an equation of the circle in standard form.
-The center is $( - 5 , - 4 )$ and another point on the circle is $( 1,4 )$
A) $( x - 5 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 100$
B) $( x + 5 ) ^ { 2 } + ( y + 4 ) ^ { 2 } = 10$
C) $( x + 5 ) ^ { 2 } + ( y + 4 ) ^ { 2 } = 100$
D) $( x - 5 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 10$

Accepted Answer

The equation of a circle in standard form is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center of the circle and $r$ is the radius. Given the center $(-5, -4)$, the equation becomes $(x + 5)^2 + (y + 4)^2 = r^2$. To find $r^2$, calculate the distance between the center $(-5, -4)$ and the point on the circle $(1, 4)$ using the distance formula: $r^2 = (1 + 5)^2 + (4 + 4)^2 = 6^2 + 8^2 = 36 + 64 = 100$. Therefore, the correct equation is $(x + 5)^2 + (y + 4)^2 = 100$.

Question 13

Use the given information about a circle to write an equation of the circle in standard form.
-The center is $( 4 , - 1 )$ and the circle is tangent to the $x$-axis.
A) $( x - 4 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 16$
B) $( x - 4 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 1$
C) $( x + 4 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 1$
D) $( x + 4 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 16$

Accepted Answer

The answer of Use the given information about a circle...

Question 14

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-The distance from the center of a circle to any point on the circle is called the and is
often denoted by r.

Accepted Answer

The answer of Write the word or phrase that best...

Question 15

Find an equation of the circle with the given characteristics.
-$( 0,12 )$ and radius $\sqrt { 6 }$
A) $x ^ { 2 } + ( y + 12 ) ^ { 2 } = \sqrt { 6 }$
B) $( y + 12 ) ^ { 2 } = 6$
C) $( y - 12 ) ^ { 2 } = \sqrt { 6 }$
D) $x ^ { 2 } + ( y - 12 ) ^ { 2 } = 6$

Accepted Answer

The general equation for a circle with center $(h, k)$ and radius $r$ is $(x - h)^2 + (y - k)^2 = r^2$. Given the center $(0, 12)$ and radius $\sqrt{6}$, the equation becomes $x^2 + (y - 12)^2 = (\sqrt{6})^2$, which simplifies to $x^2 + (y - 12)^2 = 6$, matching option D.

Question 16

Identify the center and radius of the circle.
-Identify the center and radius of the circle.
$$\left( x - \frac { 4 } { 7 } \right) ^ { 2 } + y ^ { 2 } = 12$$
A) Center $= \left( \frac { 4 } { 7 } , 0 \right) ; r = 2 \sqrt { 3 }$
B) Center $= \left( \frac { 4 } { 7 } , 0 \right) ; r = 12$
C) Center $= \left( - \frac { 4 } { 7 } , 0 \right) ; r = 2 \sqrt { 3 }$
D) Center $= \left( 0 , \frac { 4 } { 7 } \right) ; r = 12$

Accepted Answer

The equation of a circle in the form $(x - h)^2 + (y - k)^2 = r^2$ has a center at $(h, k)$ and a radius $r$. Here, $h = \frac{4}{7}$, $k = 0$, and $r^2 = 12$, so $r = \sqrt{12} = 2\sqrt{3}$.

Question 17

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-An equation of a circle written in the form $$x ^ { 2 } + y ^ { 2 } + A x + B y + C = 0$$ = 0 is called the form of an equation of a circle.

Accepted Answer

The answer of Write the word or phrase that best...

Question 18

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-The standard form of an equation of a circle with center (h, k) and radius r is given by
.

Accepted Answer

The answer of Write the word or phrase that best...

Question 19

Solve the problem.
-Write an equation of the circle that is tangent to both axes with radius $\sqrt { 3 }$ and center in Quadrant IV.
A) $( x + \sqrt { 3 } ) ^ { 2 } + ( y - \sqrt { 3 } ) ^ { 2 } = \sqrt { 3 }$
B) $( x - \sqrt { 3 } ) ^ { 2 } + ( y + \sqrt { 3 } ) ^ { 2 } = 3$
C) $( x - \sqrt { 3 } ) ^ { 2 } + ( y + \sqrt { 3 } ) ^ { 2 } = \sqrt { 3 }$
D) $( x + \sqrt { 3 } ) ^ { 2 } + ( y - \sqrt { 3 } ) ^ { 2 } = 3$

Accepted Answer

The center of the circle in Quadrant IV with a radius of $\sqrt{3}$ would be at $(\sqrt{3}, -\sqrt{3})$, since it's tangent to both axes and in Quadrant IV. The general equation of a circle is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius. Substituting $h = \sqrt{3}$, $k = -\sqrt{3}$, and $r = \sqrt{3}$, we get $(x - \sqrt{3})^2 + (y + \sqrt{3})^2 = (\sqrt{3})^2$, which simplifies to $(x - \sqrt{3})^2 + (y + \sqrt{3})^2 = 3$.

Question 20

Use the given information about a circle to find its equation.
-Center $( - 5 , - 4 )$ and diameter 10
A) $( x - 5 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 25$
B) $( x + 5 ) ^ { 2 } + ( y + 4 ) ^ { 2 } = 100$
C) $( x + 5 ) ^ { 2 } + ( y + 4 ) ^ { 2 } = 25$
D) $( x - 5 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 100$

Accepted Answer

The equation of a circle with center $(h, k)$ and radius $r$ is $(x - h)^2 + (y - k)^2 = r^2$. Given the center $(-5, -4)$ and diameter $10$, the radius is $10/2 = 5$. Thus, $r^2 = 25$, and the correct equation is $(x + 5)^2 + (y + 4)^2 = 25$.

Quiz 2: Functions and Relations

Use the given information about a circle to find its equation. -Center $( - 1 , - 5 )$ and radius $\sqrt { 3 }$

Use the given information about a circle to write an equation of the circle in standard form. Graph the circle. -Center: $( 4,2 )$ ; Radius: 4

Use the given information about a circle to find its equation. -Center $( 6 , - 7 )$ and radius of 5

Choose the one alternative that best completes the statement or answers the question. Solve the problem. - $\text { Is the point } ( - 3 , - 1 ) \text { on the circle defined by } ( x + 3 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 9$

Find an equation of the circle with the given characteristics. - $( - 6,0 )$ and radius 3

Determine the center and radius of the circle. - $( x + 2 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = 45$

Use the given information about a circle to write an equation of the circle in standard form. -The endpoints of a diameter are $( 3,11 )$ and $( - 9 , - 5 )$ .

Solve the problem. -Write an equation that represents the set of points that are 5 units from $( - 5 , - 7 )$ .

Determine the center and radius of the circle. - $( x - 4 ) ^ { 2 } + ( y - 8 ) ^ { 2 } = 9$

Find an equation of the circle with the given characteristics. -Center $( 8 , - 7 )$ and radius 3

Find an equation of the circle with the given characteristics. -Center $( - 10 , - 2 )$ and radius 6

Use the given information about a circle to write an equation of the circle in standard form. -The center is $( - 5 , - 4 )$ and another point on the circle is $( 1,4 )$

Use the given information about a circle to write an equation of the circle in standard form. -The center is $( 4 , - 1 )$ and the circle is tangent to the $x$ -axis.

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The distance from the center of a circle to any point on the circle is called the and is often denoted by r.

Find an equation of the circle with the given characteristics. - $( 0,12 )$ and radius $\sqrt { 6 }$

Identify the center and radius of the circle. -Identify the center and radius of the circle. $\left( x - \frac { 4 } { 7 } \right) ^ { 2 } + y ^ { 2 } = 12$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -An equation of a circle written in the form $x ^ { 2 } + y ^ { 2 } + A x + B y + C = 0$ = 0 is called the form of an equation of a circle.

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The standard form of an equation of a circle with center (h, k) and radius r is given by .

Solve the problem. -Write an equation of the circle that is tangent to both axes with radius $\sqrt { 3 }$ and center in Quadrant IV.

Use the given information about a circle to find its equation. -Center $( - 5 , - 4 )$ and diameter 10

Quiz 2: Functions and Relations

Use the given information about a circle to find its equation. -Center (−1,−5) ( - 1 , - 5 ) (−1,−5) and radius 3\sqrt { 3 }3​

Use the given information about a circle to write an equation of the circle in standard form. Graph the circle. -Center: (4,2) ( 4,2 ) (4,2) ; Radius: 4

Use the given information about a circle to find its equation. -Center (6,−7) ( 6 , - 7 ) (6,−7) and radius of 5

Find an equation of the circle with the given characteristics. - (−6,0) ( - 6,0 ) (−6,0) and radius 3

Determine the center and radius of the circle. - (x+2) 2+(y−3) 2=45( x + 2 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = 45(x+2) 2+(y−3) 2=45

Use the given information about a circle to write an equation of the circle in standard form. -The endpoints of a diameter are (3,11) ( 3,11 ) (3,11) and (−9,−5) ( - 9 , - 5 ) (−9,−5) .

Solve the problem. -Write an equation that represents the set of points that are 5 units from (−5,−7) ( - 5 , - 7 ) (−5,−7) .

Determine the center and radius of the circle. - (x−4) 2+(y−8) 2=9( x - 4 ) ^ { 2 } + ( y - 8 ) ^ { 2 } = 9(x−4) 2+(y−8) 2=9

Find an equation of the circle with the given characteristics. -Center (8,−7) ( 8 , - 7 ) (8,−7) and radius 3

Find an equation of the circle with the given characteristics. -Center (−10,−2) ( - 10 , - 2 ) (−10,−2) and radius 6

Use the given information about a circle to write an equation of the circle in standard form. -The center is (−5,−4) ( - 5 , - 4 ) (−5,−4) and another point on the circle is (1,4) ( 1,4 ) (1,4)

Use the given information about a circle to write an equation of the circle in standard form. -The center is (4,−1) ( 4 , - 1 ) (4,−1) and the circle is tangent to the xxx -axis.

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The distance from the center of a circle to any point on the circle is called the and is often denoted by r.

Find an equation of the circle with the given characteristics. - (0,12) ( 0,12 ) (0,12) and radius 6\sqrt { 6 }6​

Identify the center and radius of the circle. -Identify the center and radius of the circle. (x−47) 2+y2=12\left( x - \frac { 4 } { 7 } \right) ^ { 2 } + y ^ { 2 } = 12(x−74​) 2+y2=12

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -An equation of a circle written in the form x2+y2+Ax+By+C=0x ^ { 2 } + y ^ { 2 } + A x + B y + C = 0x2+y2+Ax+By+C=0 = 0 is called the form of an equation of a circle.

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The standard form of an equation of a circle with center (h, k) and radius r is given by .

Solve the problem. -Write an equation of the circle that is tangent to both axes with radius 3\sqrt { 3 }3​ and center in Quadrant IV.

Use the given information about a circle to find its equation. -Center (−5,−4) ( - 5 , - 4 ) (−5,−4) and diameter 10

Use the given information about a circle to find its equation. -Center $( - 1 , - 5 )$ and radius $\sqrt { 3 }$

Use the given information about a circle to write an equation of the circle in standard form. Graph the circle. -Center: $( 4,2 )$ ; Radius: 4

Use the given information about a circle to find its equation. -Center $( 6 , - 7 )$ and radius of 5

Find an equation of the circle with the given characteristics. - $( - 6,0 )$ and radius 3

Determine the center and radius of the circle. - $( x + 2 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = 45$

Use the given information about a circle to write an equation of the circle in standard form. -The endpoints of a diameter are $( 3,11 )$ and $( - 9 , - 5 )$ .

Solve the problem. -Write an equation that represents the set of points that are 5 units from $( - 5 , - 7 )$ .

Determine the center and radius of the circle. - $( x - 4 ) ^ { 2 } + ( y - 8 ) ^ { 2 } = 9$

Find an equation of the circle with the given characteristics. -Center $( 8 , - 7 )$ and radius 3

Find an equation of the circle with the given characteristics. -Center $( - 10 , - 2 )$ and radius 6

Use the given information about a circle to write an equation of the circle in standard form. -The center is $( - 5 , - 4 )$ and another point on the circle is $( 1,4 )$

Use the given information about a circle to write an equation of the circle in standard form. -The center is $( 4 , - 1 )$ and the circle is tangent to the $x$ -axis.

Find an equation of the circle with the given characteristics. - $( 0,12 )$ and radius $\sqrt { 6 }$

Identify the center and radius of the circle. -Identify the center and radius of the circle. $\left( x - \frac { 4 } { 7 } \right) ^ { 2 } + y ^ { 2 } = 12$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -An equation of a circle written in the form $x ^ { 2 } + y ^ { 2 } + A x + B y + C = 0$ = 0 is called the form of an equation of a circle.

Solve the problem. -Write an equation of the circle that is tangent to both axes with radius $\sqrt { 3 }$ and center in Quadrant IV.

Use the given information about a circle to find its equation. -Center $( - 5 , - 4 )$ and diameter 10