Section 1.9
Distance and Midpoint Formulas;
Circles
The Distance Formula
y


Find the D istance betw een (-4,2) and (3, -7)


 x 2  x1 
2
  y 2  y1 
2



3  4    7  2 
2

2










49  81



130  11.4











Example
Find the distance between (4,-5) and (9,-2).
The Midpoint Formula
Find the m idpoint
Find the m idpoint of the segm entw hose endpoint a
w hose 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
T o G raph a C ircle;
First S olve the equation for y: x  y  4
2
2
y  4-x
2
y = 
G raph as t w o separate equations
y1 =
2
4x
4x
2
2
y2 = 
4x
2
S o that the circle doesn't look flattene d, press ZO O M , #5 for ZS quare.
N ow press G R A P H .
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
y
( x  4)  ( y  1)  9
2
2






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 w e take the equation from the previou s
problem w e can m ultiply out the factors
and m ove all term s to one side to get th e
general form of the equation of the circ le.
( x  6)  ( y  5)  49
2
2
x  12 x  36  y  10 y  25  49
2
2
x  y  12 x  10 y  12  0
2
2
General
Form
C om plete the square and w rite the
equation in standard form . T hen
give the center and radius of each
circle.
x  y  14 x  8 y  29  0
2
2
x   14 x
2
y 8y 
2
?
?
  29
(x   14 x  49)+ ( y  8 y  16)   29  49  16
2
 x-7 
2
2
  y+ 4   36
2
C enter (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).
(a)
7

  3, 
2

(b)
9

  2, 
2

(c)
9
3
 , 
2
2
(d)
9

  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