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8/7/2019 6-1 Circles (Presentation)
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6-1 Circles
Unit 6 Conics
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Concepts and Objectives
Circles (Obj. #19)
Identify the equation of a circle. Write the equation of a circle, given the center and
the radius.
se t e comp et ng t e square met o to eterm nethe center and radius of a circle.
Write the equation of a circle, given the center and a
point on the circle.
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Graphing Circles
To graph a circle on graph paper, plot the center point
and count out the radius. Open the compass to thatpoint and draw the circle.
2 2
Center: (3, 4),
radius =
=
=9 3
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Graphing Circles
Enter
Enter
Change theq setting to square
( )= +2
9 3 4y x
( )= +29 3 4y x
The calculator may show a gap, butthats okay.
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General Form of a Circle
Suppose we took the center-radius equation and
expanded the binomials , and set everything equal to 0:
( ) ( ) + =2 2 2 x h y k r
+ + + =2 2 2 2 22 2 0 x hx h k k r
If we letc = 2h, d= 2k, and e = h2 + k2 r2, we have
( ) ( ) ( )+ + + + + =2 2 2 2 22 2 0x y h x k y h k r
+ + + + =2 2
0 x y cx dy e
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General Form of a Circle
To get from the general equation back to the center-
radius form (so we can know the center and the radius),we complete the square for bothxandy.
whose equation is
+ + =2 2 4 8 44 0 x y x y
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General Form of a Circle
Example: What is the center and radius of the circle
whose equation is+ + =2 2 4 8 44 0 x y x y
( ) ( )+ + =2 24 8 44 x x y y
The center is at(2, 4), and the radius is 8.
+ + =
+ + + +
22
22
4 8 444 8
4 162 2
x x y y
( ) ( )+ + + + =2 2 2 24 2 8 4 64 x x y y
( ) ( )+ + =2 2 2
2 4 8x y
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General Form of a Circle
Example: What is the center and radius of the circle
whose equation is+ + =2 22 2 2 6 45 0 x y x y
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General Form of a Circle
Example: What is the center and radius of the circle
whose equation is+ + =2 22 2 2 6 45 0 x y x y
( ) ( ) + + =2 22 2 3 45 x x y y
In order to complete the
square, the coefficients of
the square term must be 1.
Dont forget
to distribute!
+ + = + + + +
2 2
2 2
2 2 3 451 3 1 9
2 2 2 2 x x y y
+ + =
2 2
1 3
2 2 502 2x y
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General Form of a Circle
Example (cont.):
+ + =
2 21 3
2 2 502 2
x yDivide through
by 2.
The center is at , and the radius is 5.
+ + =
2 2
1 3 252 2
x y
1 3
,2 2
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Characteristics Example: The graph of the equation
is either a point or is nonexistent. Which is it?+ + + =
2 2
8 2 24 0 x y x y
2 2
2 28 2
Since r2 is negative, the graph is nonexistent.
2 2
( ) ( ) + + + + = 2 2 2 28 4 2 1 7 x x y y
( ) ( ) + + = 2 2
4 1 7x y
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Writing the Equation of a CircleWe can now tell that the equation
is a circle with a center at(2, 3) and a radius of 4.
Suppose I wanted to know whether (6, 3) was on the
( ) ( ) + =2 2
2 3 16x y
c rc e. ow cou n out In order to be on the circle, the point must satisfy the
equation. That is, if we plug in 6 forxand 3 fory, and
we get 16, the point is on the circle.
( ) ( ) + =2 2
6 2 3 3 16?
=16 16
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Writing the Equation of a Circle We can use this idea to write the equation of a circle
given the center and a point on the circle.
Example: Write the equation of the circle with center at
, , .
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Writing the Equation of a Circle Example: Write the equation of the circle with center at
(4, 5) that contains the point(2, 3).
( ) ( )+ + =2 2 2
4 5 x y r
=2 2 2
Therefore, the equation of the circle is
= 28 r
( ) ( )+ + =
2 2
4 5 8x y Dont square the8its already
squared!
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Circular Inequalities Circular inequalities are fairly straightforward. For the
center-radius form of the circle, the graph will be The region inside the circle if the symbol is or
s w t nes, < or > s grap e w t a otte ne an
or is graphed with a solid line.
<
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Homework College Algebra (brown book)
Page 199: 21-30 (3s) Turn in: 21, 24
Algebra & Trigonometry(green book)
Page 466: 2-18 (even) Turn in: 4, 6, 12, 14