Section 2.8
Distance and Midpoint Formulas;
Circles
The Distance Formula
y


Find the Distance between (-4,2) and (3,-7)

 x 2  x1    y2  y1 
2
 3  4    7  2 
2

2




2










49  81
130  11.4














Example
Find the distance between (4,-5) and (9,-2).
The Midpoint Formula
Find the midpoint
Find the midpoint of the segmentwhose endpoint a
whose endpoint are (-1,5) and (6,8).
 1  6 5  8 
,
(6,8)



1

6
5

8


2
2


,


2 
 2
 5 13 
(-1,5)
, 

 5 13 
2
2


,


2
2


y









x













Example
Find the midpoint of the segment whose
endpoints are (-1,7) and (-5,9).
Circles
Graphing Calculator
To Graph a Circle;
First Solve the equation for y: x 2  y 2  4
y 2  4-x 2
y =  4  x2
Graph as two separate equations
y1 = 4  x 2
y2 =  4  x2
So that the circle doesn't look flattened, press ZOOM, #5 for ZSquare.
Now press GRAPH.
Write the standard form of the equation of
the circle with center (-4,1) and radius of 3.
( x  4)  ( y  1)  3
2
2
2
Standard
Form
( x  4)  ( y  1)  9
2
2
y






3


(-4,1)













x










Find the center and radius of the circle whose
2
2
equation is ( x  3)  ( y  4)  9
Graph the equation.
Use the graph to identify the relation’s domain
and range. Why is it a relation and not a
function?
y



Center(-3,4); radius=3

3


Domain: [-6,0]; Range:[1,7]
(-3,4)












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. ( x  6)2  ( y  5) 2  49
y






x


























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.
( x  6)  ( y  5)  49
2
2
x 2  12 x  36  y 2  10 y  25  49
x  y  12 x  10 y  12  0
2
2
General
Form
Complete the square and write the
equation in standard form. Then
give the center and radius of each
circle.
x 2  y 2  14 x  8 y  29  0
x  14 x ? y  8 y  ?  29
(x 2  14 x  49)+(y 2  8 y  16)  29  49  16
2
2
 x-7    y+4 
2
2
 36
Center (7,-4); radius=6
Example
Complete the square and write the equation in
standard form. Then give the center and radius of the
circle and graph the equation. x 2  y 2  4 x  12 y  15  0
y
























x










Example
Complete the square and write the equation in
standard form. Then give the center and radius
of the circle and graph the equation. x 2  y 2  6 x  8 y  0
y
























x










Find the distance between the points (-1,8)
and (9,5).
(a)
55
(b)
73
(c)
91
(d)
109
Find the midpoint of the line segment with the
endpoints of (-1,-1) and (- 5,8).
7

 3, 
2

9

(b)  2, 
2

(a)
3 9
(c)  ,  
2 2
9

(d)  3, 2 


Write the equation of the line with
center at (0,7) and a radius of 4.
(a) x 2  y 2  49
(b) (x-7) 2  ( y  7) 2  16
(c) x 2  ( y  7) 2  16
(d) ( y  7) 2  16