Deriving The Equation Of A Circle Date Page 2

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Practice – Unit 9 (cont.)
Changing Standard Form to General Form:
(
) (
)
+
2
+
2
=
Example:
Rewrite in General Form:
.
x
2
y
7
9
(
)(
) (
)(
)
+
+
+
=
Multiply the quadratic terms:
x
2
x
2
y
7
y
7
9
+
+
+ +
+
=
2
2
x
2
x
2
x
4
y
7
y
7
y
49 9
+
+
+
=
Collect like terms:
2
2
x
4
x y
14
y
53 9
+
+
+
=
Reorder terms:
2
2
x
y
4
x
14
y
53 9
+
+
+
=
Set equal to zero:
(
)
2
2
subtract 9 from both sides
x
y
4
x
14
y
44 0
What if the equation is NOT given in Standard Form?
+
+
=
Example: Find the center and radius of a circle whose equation is
.
2
2
x
4
x y
2
y
4
(
) (
)
2
+
2
=
a) Complete the square to write the equation in the form
.
2
x h
y k
r
+
+
+
+
= +
(Set up to complete the square.)
b)
2
2
x
4
x
___
y
2
y
___
4 _______
2
2
 
4
2
+
+
+
+
= +
c)
(Add
and
to both sides.)
2
2
 
x
4
x
___
y
2
y
___
4 _______
 
2
2
+
+
+
+
= +
d)
(Simplify.)
2
2
x
4
x
___
y
2
y
___
4 _______
(
) (
)
2
+
+
2
=
e)
(Factor.)
x
___
y
___
______
f) Identify h, k, and r to determine the center and radius.
h = _______, k = _______, r = _______
So, the center is (____, ____) and the radius is _____.
Translations in the Plane:
(
) (
)
+
Suppose you translate any circle by the translation
.
x y
,
x
3,
y
2
Describe what happens to each:
a) The graph of the circle. ______________________________________
b) h ______________________________________
c) k ______________________________________
d) r ______________________________________
Finding the radius:
(
) (
)
+
2
+
2
=
A) Given the Standard Form equation:
.
x
1
y
2
9
Radius =
= 3 (
)
square root of the constant
9
+
+
+ =
B) Given General Form:
.
2
2
x
y
6
x
8 0
Must be changed into Standard Form… (
complete the square
)
+
+ +
+
+ + = + +
2
2
x
6
x
_
y
0
y
_ 8 0 _ _
+
+ +
+
+ + = + +
2
2
x
6
x
9
y
0
y
0
8 0
9
0
(
) (
)
+
2
+
+
2
+ =
x
3
y
0
8 9
(
) (
)
+
+
+
=
2
2
x
3
y
0
1
Radius =
= 1 (
)
square root of the constant
1
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