# 11.2E: Exercises

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### Practice Makes Perfect

##### Exercise $$\PageIndex{23}$$ Use the Distance Formula

In the following exercises, find the distance between the points. Write the answer in exact form and then find the decimal approximation, rounded to the nearest tenth if needed.

1. $$(2,0)$$ and $$(5,4)$$
2. $$(-4,-3)$$ and $$(2,5)$$
3. $$(-4,-3)$$ and $$(8,2)$$
4. $$(-7,-3)$$ and $$(8,5)$$
5. $$(-1,4)$$ and $$(2,0)$$
6. $$(-1,3)$$ and $$(5,-5)$$
7. $$(1,-4)$$ and $$(6,8)$$
8. $$(-8,-2)$$ and $$(7,6)$$
9. $$(-3,-5)$$ and $$(0,1)$$
10. $$(-1,-2)$$ and $$(-3,4)$$
11. $$(3,-1)$$ and $$(1,7)$$
12. $$(-4,-5)$$ and $$(7,4)$$

1. $$d=5$$

3. $$13$$

5. $$5$$

7. $$13$$

9. $$d=3 \sqrt{5}, d \approx 6.7$$

11. $$d=\sqrt{68}, d \approx 8.2$$

##### Exercise $$\PageIndex{24}$$ Use the Midpoint Formula

In the following exercises,

1. find the midpoint of the line segments whose endpoints are given and
2. plot the endpoints and the midpoint on a rectangular coordinate system.
1. $$(0,-5)$$ and $$(4,-3)$$
2. $$(-2,-6)$$ and $$(6,-2)$$
3. $$(3,-1)$$ and $$(4,-2)$$
4. $$(-3,-3)$$ and $$(6,-1)$$

1.

1. Midpoint: $$(2,-4)$$

3.

1. Midpoint: $$\left(3 \frac{1}{2},-1 \frac{1}{2}\right)$$
##### Exercise $$\PageIndex{25}$$ Write the Equation of a Circle in Standard Form

In the following exercises, write the standard form of the equation of the circle with the given radius and center $$(0,0)$$.

1. Radius: $$7$$
2. Radius: $$9$$
3. Radius: $$\sqrt{2}$$
4. Radius: $$\sqrt{5}$$

1. $$x^{2}+y^{2}=49$$

3. $$x^{2}+y^{2}=2$$

##### Exercise $$\PageIndex{26}$$ Write the Equation of a Circle in Standard Form

In the following exercises, write the standard form of the equation of the circle with the given radius and center

1. Radius: $$1$$, center: $$(3,5)$$
2. Radius: $$10$$, center: $$(-2,6)$$
3. Radius: $$2.5$$, center: $$(1.5, -3.5)$$
4. Radius: $$1.5$$, center: $$(-5.5, -6.5)$$

1. $$(x-3)^{2}+(y-5)^{2}=1$$

3. $$(x-1.5)^{2}+(y+3.5)^{2}=6.25$$

##### Exercise $$\PageIndex{27}$$ Write the Equation of a Circle in Standard Form

For the following exercises, write the standard form of the equation of the circle with the given center with point on the circle.

1. Center $$(3,−2)$$ with point $$(3,6)$$
2. Center $$(6,−6)$$ with point $$(2,−3)$$
3. Center $$(4,4)$$ with point $$(2,2)$$
4. Center $$(−5,6)$$ with point $$(−2,3)$$

1. $$(x-3)^{2}+(y+2)^{2}=64$$

3. $$(x-4)^{2}+(y-4)^{2}=8$$

##### Exercise $$\PageIndex{28}$$ Graph a Circle

In the following exercises,

1. find the center and radius, then
2. graph each circle.
1. $$(x+5)^{2}+(y+3)^{2}=1$$
2. $$(x-2)^{2}+(y-3)^{2}=9$$
3. $$(x-4)^{2}+(y+2)^{2}=16$$
4. $$(x+2)^{2}+(y-5)^{2}=4$$
5. $$x^{2}+(y+2)^{2}=25$$
6. $$(x-1)^{2}+y^{2}=36$$
7. $$(x-1.5)^{2}+(y+2.5)^{2}=0.25$$
8. $$(x-1)^{2}+(y-3)^{2}=\frac{9}{4}$$
9. $$x^{2}+y^{2}=64$$
10. $$x^{2}+y^{2}=49$$
11. $$2 x^{2}+2 y^{2}=8$$
12. $$6 x^{2}+6 y^{2}=216$$

1.

1. The circle is centered at $$(−5,−3)$$ with a radius of $$1$$.

3.

1. The circle is centered at $$(4,−2)$$ with a radius of $$4$$.

5.

1. The circle is centered at $$(0,−2)$$ with a radius of $$5$$.

7.

1. The circle is centered at $$(1.5,2.5)$$ with a radius of $$0.5$$.

9.

1. The circle is centered at $$(0,0)$$ with a radius of $$8$$.

11.

1. The circle is centered at $$(0,0)$$ with a radius of $$2$$.
##### Exercise $$\PageIndex{29}$$ Graph a Circle

In the following exercises,

1. identify the center and radius and
2. graph.
1. $$x^{2}+y^{2}+2 x+6 y+9=0$$
2. $$x^{2}+y^{2}-6 x-8 y=0$$
3. $$x^{2}+y^{2}-4 x+10 y-7=0$$
4. $$x^{2}+y^{2}+12 x-14 y+21=0$$
5. $$x^{2}+y^{2}+6 y+5=0$$
6. $$x^{2}+y^{2}-10 y=0$$
7. $$x^{2}+y^{2}+4 x=0$$
8. $$x^{2}+y^{2}-14 x+13=0$$

1.

1. Center: $$(−1,−3)$$, radius: $$1$$

3.

1. Center: $$(2,−5)$$, radius: $$6$$

5.

1. Center: $$(0,−3)$$, radius: $$2$$

7.

1. Center: $$(−2,0)$$, radius: $$-2$$
##### Exercise $$\PageIndex{30}$$ Writing Exercises
1. Explain the relationship between the distance formula and the equation of a circle.
2. Is a circle a function? Explain why or why not.
3. In your own words, state the definition of a circle.
4. In your own words, explain the steps you would take to change the general form of the equation of a circle to the standard form.

## Self Check

a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.