Section 2.8Distance and Midpoint Formulas and Circles

1Section 2.8 Distance and Midpoint Formulas and Circles

The Distance Formula

2Section 2.8 Distance and Midpoint Formulas and Circles

3Section 2.8 Distance and Midpoint Formulas and Circles

Figure 2.59

Distance Formula

2 22 1 2 1

2 2

Find the distance between (-4,2) and (3,-7).

x

3 4 7 2

49 81

130 11.4

x y y

x

y

4

Section 2.8 Distance and Midpoint Formulas and Circles

Example

Find the distance between (4,-5) and (9,-2).

5Section 2.8 Distance and Midpoint Formulas and Circles

The Midpoint Formula

6Section 2.8 Distance and Midpoint Formulas and Circles

Find the midpoint of the segmentwhose endpoints are (-1,5) and (6,8).

1 6 5 8,2 2

5 13,2 2

x

y

Find the midpoint of the segmentwhose endpoint are (-1,5) and (6,8).

1 6 5 8,2 2

5 13,2 2

(-1,5)

(6,8)

7Section 2.8 Distance and Midpoint Formulas and Circles

Example

Find the midpoint of the segment whose endpoints are (-1,7) and (-5,9).

8Section 2.8 Distance and Midpoint Formulas and Circles

Circles

9Section 2.8 Distance and Midpoint Formulas and Circles

10Section 2.8 Distance and Midpoint Formulas and Circles

Write the standard form of the equation of the circle with the center (-4,1) and a radius of 3.

x

y

2 2 2

2 2

( 4) ( 1) 3

( 4) ( 1) 9

x y

x y

Standard Form

(-4,1)3

11Section 2.8 Distance and Midpoint Formulas and Circles

Find the center and radius of the circle whose equation isGraph the equation.Use the graph to identify the relation’s domain and range. Why is it a relation and not a function?

2 2( 3) ( 4) 9x y

x

y

3

(-3,4)Domain: [-6,0]; Range: [1,7]

Center (-3,4); radius = 3

12Section 2.8 Distance and Midpoint Formulas and Circles

Example

Write the standard form of the equation of a circle with the center at (-2,7) and a radius of 5.

13Section 2.8 Distance and Midpoint Formulas and Circles

ExampleFind the center and radius of the circle whose equation is below. Graph the equation. Use the graph to identify the relation’s domain and range. 2 2( 6) ( 5) 49x y

x

y

14Section 2.8 Distance and Midpoint Formulas and Circles

2 2

2 2

If we take the equation from the previousproblem we can multiply out the factorsand move all terms to one side to get the general form of the equation of the circle.

( 6) ( 5) 49

12 36 10 25

x y

x x y y

2 2

49

12 10 12 0x y x y

General Form

15Section 2.8 Distance and Midpoint Formulas and Circles

2 2

2 2

2 2

2 2

Complete the square and write the equation in standard form. Thengive the center and radius of each circle.

14 8 29 0

14 8 29

( 14 49) ( 8 16) 29 49 16

- 7 4

x y x y

x x y y

x x y y

x y 36

(7, -4); radius=6Center

? ?

16Section 2.8 Distance and Midpoint Formulas and Circles

Example

2 2x 4 12 15 0y x y

x

y

Complete the square and write the equation in standard form. Then give the center and radius of the circle and graph the equation.

17Section 2.8 Distance and Midpoint Formulas and Circles

Example

2 2 6 8 0 x y x y

x

y

Complete the square and write the equation in standard form. Then give the center and radius of the circle and graph the equation.

18Section 2.8 Distance and Midpoint Formulas and Circles

(a)

(b)

(c)

(d)

Find the distance between the points (-1,8) and (9,5).

55

73

91

109

19Section 2.8 Distance and Midpoint Formulas and Circles

(a)

(b)

(c)

(d)

Find the midpoint of the line segment with the endpoints of (-1,-1) and (- 5,8).

73,292,2

3 9,2 2

93,2

20Section 2.8 Distance and Midpoint Formulas and Circles

(a)

(b)

(c)

(d)

Write the equation of the circle with the center at (0,7) and a radius of 4.

2 2

2 2

2 2

2

49

( - 7) ( 7) 16

( 7) 16

( 7) 16

x y

x y

x y

y

21Section 2.8 Distance and Midpoint Formulas and Circles