Question

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equation of the circle hw
in exercises 1–4, write the standard equation of the circle with the given center and radius.

1. center: (0, 0), radius: 8
2. center: (0, 5), radius: 2
3. center: (10, 5), radius: $sqrt{3}$
4. center: $(-3, 2)$, radius: $sqrt{10}$
5. write the standard equation for each circle in the diagram at the right.

a. $odot a$
b. $odot b$
c. $odot c$
in exercises 6–8, match each graph with its equation.
6.
7.
8.
a. $x^2 + y^2 = 4$
b. $(x - 3)^2 + y^2 = 4$
c. $(x + 3)^2 + y^2 = 4$

Explanation:

The standard equation of a circle is $(x-h)^2 + (y-k)^2 = r^2$, where $(h,k)$ is the center and $r$ is the radius.

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Exercises 1-4

Step1: Apply circle standard formula

For center $(h,k)$ and radius $r$, use $(x-h)^2+(y-k)^2=r^2$.

1. Center $(0,0)$, Radius 8

Substitute $h=0,k=0,r=8$:
$(x-0)^2 + (y-0)^2 = 8^2$
Simplify: $x^2 + y^2 = 64$

2. Center $(0,5)$, Radius 2

Substitute $h=0,k=5,r=2$:
$(x-0)^2 + (y-5)^2 = 2^2$
Simplify: $x^2 + (y-5)^2 = 4$

3. Center $(10,5)$, Radius $\sqrt{3}$

Substitute $h=10,k=5,r=\sqrt{3}$:
$(x-10)^2 + (y-5)^2 = (\sqrt{3})^2$
Simplify: $(x-10)^2 + (y-5)^2 = 3$

4. Center $(-3,2)$, Radius $\sqrt{10}$

Substitute $h=-3,k=2,r=\sqrt{10}$:
$(x-(-3))^2 + (y-2)^2 = (\sqrt{10})^2$
Simplify: $(x+3)^2 + (y-2)^2 = 10$

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Exercise 5

Step1: Identify center/radius from graph

For each circle, read $(h,k)$ and $r$ from grid.

A. $\odot A$

Center $(-4,-4)$, Radius 2:
$(x-(-4))^2 + (y-(-4))^2 = 2^2$
Simplify: $(x+4)^2 + (y+4)^2 = 4$

B. $\odot B$

Center $(-1,2)$, Radius 3:
$(x-(-1))^2 + (y-2)^2 = 3^2$
Simplify: $(x+1)^2 + (y-2)^2 = 9$

C. $\odot C$

Center $(3,-2)$, Radius 3:
$(x-3)^2 + (y-(-2))^2 = 3^2$
Simplify: $(x-3)^2 + (y+2)^2 = 9$

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Exercises 6-8

Step1: Match graph to equation

Identify center from graph, compare to $(x-h)^2+(y-k)^2=r^2$.

6. Graph

Center $(3,0)$, Radius 2: Matches $(x-3)^2 + y^2 = 4$

7. Graph

Center $(0,0)$, Radius 2: Matches $x^2 + y^2 = 4$

8. Graph

Center $(-3,0)$, Radius 2: Matches $(x+3)^2 + y^2 = 4$

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Answer:

Exercises 1-4

1. $x^2 + y^2 = 64$
2. $x^2 + (y-5)^2 = 4$
3. $(x-10)^2 + (y-5)^2 = 3$
4. $(x+3)^2 + (y-2)^2 = 10$

Exercise 5

A. $(x+4)^2 + (y+4)^2 = 4$
B. $(x+1)^2 + (y-2)^2 = 9$
C. $(x-3)^2 + (y+2)^2 = 9$

Exercises 6-8

1. B. $(x-3)^2 + y^2 = 4$
2. A. $x^2 + y^2 = 4$
3. C. $(x+3)^2 + y^2 = 4$