Transcript 1.9
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
4x
4x
2
2
y2 =
4x
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