Math 1314
2.8 Distance and Midpoint Formulas-Circles
Distance Formula: Used to find the distance between two points x1, y1  and x2 , y2 
distance  ( x2  x1 ) 2  ( y2  y1 ) 2
Example 1: Find the distance between A(4,8) and B(1,12)
distance  ( x2  x1 ) 2  ( y2  y1 ) 2
distance  (1  4) 2  (12  8) 2
distance  (3) 2  (4) 2
distance  9  16  25  5
Example 2: Find the distance between:
a/ A(2, 7) and B(11, 9)
b/ C(-5, 8) and D(2, - 4)
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Section 2.8 Notes
Math 1314

Section 2.8 Notes


c/ M 3 3 , 5 and N  3, 4 5

Midpoint Formula: Used to find the center of a line segment Ax1 , y1  and B  x2 , y2 
x x y y 
midpoint   2 1 , 2 1 
2 
 2
Example 3: Find the midpoint between A(4,8) and B(1,12)
x x y y 
midpoint   2 1 , 2 1 
2 
 2
 1  4 12  8 
midpoint  
,

2 
 2
midpoint   5 ,10 


2

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Math 1314
Section 2.8 Notes
Example 4: Find the midpoint between:
a/ A(2, 7) and B(14, 9)
b/ C(-5, 8) and D(2, - 4)



c/ M 18 ,  4 and N 2 , 4

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Math 1314
Circles
Section 2.8 Notes
Definition of Circle: A circle is a set of points in a plane that are located a fixed distance, called the radius
from a given point in the plane called the center.
Standard Form of the Equation of a Circle:
The standard form of the equation of a circle with the center (h, k) and the radius r is (x – h)2 + (y – k)2 = r2
Example 1: State the center and radius of each circle then graph it.
a/ (x – 5)2 + (y + 3)2 = 4
b/ (x + 7)2 + (y – 2)2 = 1
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Math 1314
Example 2: Write the equation of the circle in standard form given:
a/ Center C(3, -5) and radius r = 7.
b/ Center C(0, 2) and passing through (3, -1).
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Section 2.8 Notes
Math 1314
Section 2.8 Notes
General Form:
The equation of a circle can be written as x2 + y2 + ax + by + c = 0, this is called general form.
Remark: Equation x2 + y2 + ax + by + c = 0 does not always represent a circle. It may represent a point or no
graph.
Example 2:
Write the equation of the circle in standard form and general form given: Center (-3, 8)and radius is 6.
Since the center is (-3, 8), we have h = - 3, k = 8 and the radius is 6, so r = 6.
Then the equation is (x + 3)2 + (y – 8)2 = 36 (standard form)
And expand the squares and simplify to obtain the general form x2 + y2 + 6x – 16y + 37 = 0.
Example 3:
Write the equation of the circle in standard form and general form given: Center (2, -9) and radius is 11 .
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Math 1314
Find Center and Radius of a Circle Given a General Equation
We need to write the equation of a circle in a standard form.
Example 4: Find center and radius of a circle whose equation is
a/ x 2  y 2  10 x  2 y  17  0
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Section 2.8 Notes
Math 1314
b/ x 2  y 2  4 x  8 y  20  0
2
2
c/ x  y  4 x  8 y  25  0
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Section 2.8 Notes