The Distance Formula and the Circle

(1)

767

and the Circle

10.2

OBJECTIVES

1. Given a center and radius, find the equation of a circle

2. Given an equation for a circle, find the center and radius

3. Given an equation, sketch the graph of a circle

In Section 10.1, we examined the parabola. In this section, we turn our attention to another conic section, the circle.

The distance formula is central to any discussion of conic sections.

Equation (1) can be used in two ways. Given the center and radius of the circle, we can write its equation; or given its equation, we can find the center and radius of a circle.

We can use the distance formula to derive the algebraic equation of a circle, given its center and its radius.

Suppose a circle has its center at a point with coordinates (h, k) and radius r. If (x, y) rep- resents any point on the circle, then, by its definition, the distance from (h, k) to (x, y) is r.

Applying the distance formula, we have

Squaring both sides of the equation gives the equation of the circle r

²

(x h)

²

(y k)

²

In general, we can write the following equation of a circle.

r 2(x h)

²

(y k)

²

A circle is the set of all points in the plane equidistant from a fixed point, called the center of the circle. The distance between the center of the circle and any point on the circle is called the radius of the circle.

Definitions: Circle

The distance, d, between two points (x₁, y₁) and (x₂, y₂) is given by d 2(x2 x1)² (y2 y1)²

Definitions: The Distance Formula

The equation of a circle with center (h, k) and radius r is

(x h)² (y k)² r² (1)

Definitions: Equation of a Circle

NOTEA special case is the circle

centered at the origin with radius r. Then (h, k) (0, 0), and its equation is

x² y² r²

(x₂, y₂)

(x₂, y₁) (x₁, y₁)

d y

x

(x, y)

(h, k) y

x r

(2)

768 CHAPTER10 GRAPHS OFCONICSECTIONS

Finding the Center and Radius of a Circle

Find the center and radius of the circle with equation (x 1)

²

(y 2)

²

9 Remember, the general form is (x h)

²

(y k)

²

r

²

Our equation “fits” this form when it is written as

Note: y 2 y (2)

(x 1)

²

[y (2)]

²

3

²

So the center is at (1, 2), and the radius is 3. The graph is shown.

y

x 3

(1, 2)

(x 1)² (y 2)² 9

Example 2

Finding the Equation of a Circle

Find the equation of a circle with center at (2, 1) and radius 3. Sketch the circle.

Let (h, k) (2, 1) and r 3. Applying equation (1) yields (x 2)

²

[y (1)]

²

3

²

(x 2)

²

(y 1)

²

9 To sketch the circle, we locate the center of the circle. Then we determine four points 3 units to the right and left and up and down from the center of the circle. Drawing a smooth curve through those four points completes the graph.

Example 1

Now, given an equation for a circle, we can also find the radius and center and then sketch the circle. We start with an equation in the special form of equation (1).

C H E C K Y O U R S E L F 1

Find the equation of the circle with center at (2, 1) and radius 5. Sketch the circle.

y

x 3

(2, 1)

(x 2)² (y 1)² 9

NOTEThe circle can be graphed on the calculator by solving for y, then graphing both the upper half and lower half of the circle. In this case, (x 1)² (y 2)² 9 ( y 2)² 9 (x 1)²

Now graph the two functions

and

on your calculator. (The display screen may need to be squared to obtain the shape of a circle.) y 2 29 (x 1)² y 2 29 (x 1)² y 2 29 (x 1)² ( y 2) 29 (x 1)²

(3)

Finding the Center and Radius of a Circle

Find the center and radius of the circle with equation x

²

2x y

²

6y 1

Then sketch the circle.

We could, of course, simply substitute values of x and try to find the corresponding values for y. A much better approach is to rewrite the original equation so that it matches the standard form.

First, add 1 to both sides to complete the square in x.

x

²

2x 1 y

²

6y 1 1

Then add 9 to both sides to complete the square in y.

x

²

2x 1 y

²

6y 9 1 1 9

We can factor the two trinomials on the left (they are both perfect squares) and simplify on the right.

(x 1)

²

(y 3)

²

9 The equation is now in standard form, and we can see that the center is at ( 1, 3) and the radius is 3. The sketch of the circle is shown. Note the “translation” of the center to ( 1, 3).

Example 3

NOTETo recognize the

equation as having the form of a circle, note that the

coefficients of x²and y²are equal.

NOTEThe linear terms in x and y show a translation of the center away from the origin.

To graph the equation of a circle that is not in standard form, we complete the square.

Let’s see how completing the square can be used in graphing the equation of a circle.

C H E C K Y O U R S E L F 2

Find the center and radius of the circle with equation

(x 3)

²

(y 2)

²

16

Sketch the circle.

C H E C K Y O U R S E L F 3

Find the center and radius of the circle with equation

x

²

4x y

²

2y 1

Sketch the circle.

3

(x 1)² (y 3)² 9 y

x (1, 3)

(4)

770 CHAPTER10 GRAPHS OFCONICSECTIONS

C H E C K Y O U R S E L F A N S W E R S

1. (x 2)

²

(y 1)

²

25 2. center: ( 3, 2); radius 4

3. (x 2)

²

(y 1)

²

4; center: (2, 1); radius 2

y

(2, 1) x 2

y

(3, 2)

x 4

y

(2, 1)

x 5

(5)

Exercises

In exercises 1 to 12, decide whether each equation has as its graph a line, a parabola, a circle, or none of these.

1.

y x

²

2x 5

^2.

y

²

x

²

64

3.

y 3x 2

^4.

2y 3x 12

5.

(x 3)

²

(y 2)

²

10

^6.

y 2(x 3)

²

5

7.

x

²

4x y

²

6y 3

^8.

4x 3

9.

y

²

4x

²

36

^10.

x

²

(y 3)

²

9

11.

y 2x

²

8x 3

^12.

2x

²

3y

²

6y 13

In exercises 13 to 20, find the center and the radius for each circle.

13.

x

²

y

²

25

^14.

x

²

y

²

72

15.

(x 3)

²

(y 1)

²

16

^16.

(x 3)

²

y

²

81

17.

x

²

2x y

²

15

^18.

x

²

y

²

6y 72

19.

x

²

6x y

²

8y 16

^20.

x

²

5x y

²

3y 8

In exercises 21 to 32, graph each circle by finding the center and the radius.

21.

x

²

y

²

4

^22.

x

²

y

²

25

y

x y

x

10.2

Section Date

ANSWERS

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

771

(6)

23.

4x

²

4y

²

36

^24.

9x

²

9y

²

144

25.

(x 1)

²

y

²

9

^26.

x

²

(y 2)

²

16

27.

(x 4)

²

(y 1)

²

16

28.

(x 3)

²

(y 2)

²

25

y

x y

x

y

x y

x

y

x y

x

ANSWERS

23.

24.

25.

26.

27.

28.

772

(7)

29.

x

²

y

²

4y 12

^30.

x

²

6x y

²

0

31.

x

²

4x y

²

2y 1

32.

x

²

2x y

²

6y 6

33.

Describe the graph of x

²

y

²

2x 4y 5 0.

34.

Describe how completing the square is used in graphing circles.

35.

A solar oven is constructed in the shape of a hemisphere. If the equation x

²

y

²

500 1000

describes the outer edge of the oven in centimeters, what is its radius?

36.

A solar oven in the shape of a hemisphere is to have a diameter of 80 cm. Write the equation that describes the outer edge of this oven.

37.

A solar water heater is constructed in the shape of a half cylinder, with the water supply pipe at its center. If the water heater has a diameter of m, what is the equation that describes its outer edge?

4 3

y

x y

x

y

x y

x

29.

30.

31.

32.

33.

34.

35.

36.

37.

773

(8)

A solar water heater is constructed in the shape of a half cylinder with circumference

described by the equation 9x

²

9y

²

16 0

What is its diameter if the units for the equation are meters?

A circle can be graphed on a calculator by plotting the upper and lower semicircles on the same axes. For example, to graph x

²

y

²

16, we solve for y:

This is then graphed as two separate functions, and

In exercises 39 to 42, use that technique to graph each circle.

39.

x

²

y

²

36

40.

(x 3)

²

y

²

9

41.

(x 5)

²

y

²

36 Y

₂

216 x

²

Y

₁

216 x

²

y 216 x

²

ANSWERS

38.

39.

40.

41.

774

(9)

42.

(x 2)

²

(y 1)

²

25 Each of the following equations defines a relation. Write the domain and the range of each relation.

43.

(x 3)

²

(y 2)

²

16

^44.

(x 1)

²

(y 5)

²

9

45.

x

²

(y 3)

²

25

^46.

(x 2)

²

y

²

36 Answers

1.

Parabola

3.

Line

5.

Circle

7.

Circle

9.

None of these

11.

Parabola

13.

Center: (0, 0); radius: 5

15.

Center: (3, 1); radius: 4

17.

Center: ( 1, 0); radius: 4

^19.

Center: (3, 4); radius:

21.

x

²

y

²

4

^23.

4x

²

4y

²

36 x

²

y

²

9 Center: (0, 0); radius: 2 Center: (0, 0); radius: 3

25.

(x 1)

²

y

²

9

^27.

(x 4)

²

(y 1)

²

16 Center: (1, 0); radius: 3 Center: (4, 1); radius: 4

(x 4)² (y 1)² 16 Center: (4, 1);

radius: 4 y

x (x 1)² y² 9

Center: (1, 0);

radius: 3 y

x

4x² 4y² 36 x² y² 9 Center: (0, 0);

radius: 3 y

x x² y² 4

Center: (0, 0);

radius: 2 y

x

141

42.

43.

44.

45.

46.

775

(10)

x

²

y

²

4y 12

^31.

x

²

4x y

²

2y 1

Center: (0, 2); radius: 4 Center: (2, 1); radius: 2

33.

Circle with radius 0; center at (1, 2)

^35.

24.4 cm

37.

x

²

y

²

39.

x

²

y

²

36

41.

(x 5)

²

y

²

36

43.

Domain: x7 x 1

^45.

Domain: x5 x 5

Range: y2 y 6 Range: y2 y 8

y y

36 (x 5)² 36 (x 5)² y y

36 x² 36 x²

4 9

1500 1015 cm

x² 4x y² 2y 1 (x 2)² (y 1)² 4 Center: (2, 1);

radius: 2 y

x x² y² 4y 12

x² (y 2)² 16 Center: (0, 2);

radius: 4 y

x

776