Download Chapter 1, Section 9

Chapter 1
Functions and Graphs
1.9 Distance and Midpoint
Formulas; Circles
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
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Objectives:
•
•
•
•
•
Find the distance between two points.
Find the midpoint of a line segment.
Write the standard form of a circle’s equation.
Give the center and radius of a circle whose equation is
in standard form.
Convert the general form of a circle’s equation to
standard form.
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The Distance Formula
The distance, d, between the points (x1, y1) and (x2, y2)
in the rectangular coordinate system is
d  ( x2  x1 )2  ( y2  y1 ) 2
To compute the distance between two points, find the
square of the difference between the x-coordinates plus
the square of the difference between the y-coordinates.
The principal square root of this sum is the distance.
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Example: Using the Distance Formula
Find the distance between (–1, –3) and (2, 3). Express
the answer in simplified radical form and then round to
two decimal places.
d  ( x2  x1 )2  ( y2  y1 ) 2
 (2  (1))2  (3  (3)) 2
 (3)2  (6)2
 9  36
 45  9 5  3 5  6.71
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Find the distance. Express the answer in simplified radical
form and then round to two decimal places.
1) (-2,-6) & (3,-4)
2) (0,-2,) & (4,3)
29
41
3) ( 3 3, 5) & ( 3,4 5)
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The Midpoint Formula
Consider a line segment whose endpoints are (x1, y1)
and (x2, y2). The coordinates of the segment’s midpoint
are
 x1  x2 , y1  y2 


2 
 2
To find the midpoint, take the average of the two
x-coordinates and the average of the two y-coordinates.
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Example: Using the Midpoint Formula
Find the midpoint of the line segment with endpoints
(1, 2) and (7, –3).
 x1  x2 , y1  y2    1  7 , 2  (3) 

 

2   2
2 
 2
8 1 

 , 
2 2 
1

  4,  
2

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Find the midpoint of the line segment with endpoints
 4, 5
1)(-3, -4) & (6, –8).
2)
3)
 2 1 
 , 
 5 10 
4
 2 7
 2

,
&

,





 5 15 
 5 15 

 
7 3, 6 & 3 3, 2

5
3, 4

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Definition of a Circle
A circle is the set of all points in a plane that are equidistant
from a fixed point, called the center. The fixed distance
from the circle’s center to any point on the circle is called the
radius.
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The Standard Form of the Equation of a Circle
The standard form of the equation of a circle with center
(h, k) and radius r is
( x  h) 2  ( y  k ) 2  r 2
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Example: Finding the Standard Form of a Circle’s
Equation
Write the standard form of the equation of the circle
with center (0, –6) and radius 10.
( x  h) 2  ( y  k ) 2  r 2
( x  0)2  ( y  (6)) 2  (10) 2
x 2  ( y  6)2  100
The standard form of the equation of the circle is
x 2  ( y  6)2  100.
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Example: Using the Standard Form of a Circle’s Equation
to Graph the Circle
Find the center and the radius of the circle whose
equation is
( x  3)2  ( y  1)2  4
The center is (–3, 1).
The radius is
4  2.
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Example: Using the Standard Form of a Circle’s Equation
to Graph the Circle
Find the center and the radius of the circle whose equation
2
2
is ( x  3)  ( y  1)  4. Graph the equation.
The center is (–3, 1)

The radius is
42
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Example: Using the Standard Form of a Circle’s Equation
to Graph the Circle
Find the center and the radius of the circle whose equation
2
2
is ( x  3)  ( y  1)  4. Use the graph of the equation to
identify the relation’s domain and range.
The center is (–3, 1).
The radius is 4  2.
The domain is [–5, –1].
The range is [–1, 3].
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The General Form of the Equation of a Circle
The general form of the equation of a circle is
x 2  y 2  Dx  Ey  F  0
where D, E, and F are real numbers.
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Example: Converting the General Form of a Circle’s
Equation to Standard Form and Graphing the Circle
Write in standard form:
x2  y 2  4 x  4 y  1  0
( x 2  4 x)  ( y 2  4 y )  1
( x 2  4 x  4)  ( y 2  4 y  4)  1  4  4
( x  2)2  ( y  2) 2  9

The center of the circle is (–2, 2)
The radius of the circle is 9  3
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Example: Write equation of a circle
Write equation of a circle whose center at (3,-4) and
passing through the point (-1,2)
( x  3)2  ( y  4) 2  52
The center of the circle is (3, -4)
The radius of the circle is 52  7.2
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Example: Write equation of a circle
Write equation of a circle with diameter between (-2,3)
and (4,-5).
The center of the circle is (1,-1)
The radius of the circle is 5
( x  1)2  ( y  1) 2  25
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