Section 2.8Distance 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

(-1,5)

(6,8)

Example

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

Circles

2 2

2 2

2

To Graph a Circle;

First Solve the equation for y: x 4

y 4-x

y = 4

Graph as t

y

x

2 2

1 2wo separate equations y = 4 y = 4

So that the circle doesn't look flattened, press ZOOM, #5 for ZSquare.

Now press GRAPH.

x x

Graphing Calculator

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 line 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