Section 1.9

Distance and Midpoint Formulas;

Circles

The Distance Formula

2 2

2 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

Example

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

The Midpoint Formula

Find the midpoint of the segment

whose endpoint are (-1,5) and (6,8).

1 6 5 8,

2 2

5 13,

2 2

x

y

Find the midpoint of the segment

whose endpoint are (-1,5) and (6,8).

1 6 5 8,

2 2

5 13,

2 2

(-1,5)

(6,8)

Example

Find the midpoint of the segment whose

endpoints are (-1,7) and (-5,9).

Circles

Write the standard form of the equation of

the circle with center (-4,1) and 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

Find the center and radius of the circle whose

equation is

Graph 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

Example

Write the standard form of the equation of the

circle with center at (-2,7) and a radius of 5.

Example

Find 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

2 2

2 2

If we take the equation from the previous

problem we can multiply out the factors

and 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

2 2

2 2

2 2

2 2

Complete the square and write the

equation in standard form. Then

give the center and radius of each

circle.

x 14 8 29 0

x 14 8 29

(x 14 49)+( 8 16) 29 49 16

x-7 y+4

y x y

x y y

x y y

36

(7,-4); radius=6Center

? ?

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.

Example

2 2x 6 8 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.

(a)

(b)

(c)

(d)

Find the distance between the points (-1,8)

and (9,5).

55

73

91

109

(a)

(b)

(c)

(d)

Find the midpoint of the line segment with the

endpoints of (-1,-1) and (- 5,8).

73,

2

92,

2

3 9,

2 2

93,

2

(a)

(b)

(c)

(d)

Write the equation of the circle with

center at (0,7) and a radius of 4.

2 2

2 2

2 2

2

x 49

(x-7) ( 7) 16

x ( 7) 16

( 7) 16

y

y

y

y