Deriving The Equation Of A Circle Date
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1
2
3Geometry – Unit 9 Practice
Name: _____________________________!
Deriving the Equation of a Circle
Date: ___________ Pd: ____
G.GPE.A.1
By placing a circle in the coordinate plane with center C(h, k) and radius (r) we can derive the equation a circle.
Chose a point on the circle, P(x, y) and create a right triangle with
as the hypotenuse.
CP
1) Find the coordinates of point A. (____, ____)
2) Write an expression for the lengths of each of the legs of ∆CAP.
CA = ________
PA = ________
3) Use the Pythagorean Theorem to write an expression that represents
the length of
.
CP
2
CP
= ______________________________
CP = ______________________________
4) Another name for
is _____.
CP
5) Rewrite your expression in Part 3, substituting your answer to Part 4 for CP. _________________________
6) Square both sides.
_________________________ (Standard Form)
7) Suppose a circle has its center at the origin. What is the Standard Form equation of the circle in this case?
_________________________ (
this is considered a special case
)
General Form :
+
+
+
+ =
, where a, b, c are constants, c cannot be equal to 0 while a and b can be.
2
2
x
y
ax by c
0
8) Suppose a circle has its center at the origin. What is the General Form equation of the circle in this case?
_________________________ (
)
this is considered a special case
An equation in General Form must first be solved using the method of Completing the Squares to express it in Standard
Form. This allows you to find the center and radius.
Characteristics of the Equation of a Circle
there is always an x and a y to the second power
•
the x and y to the second power must have the same sign when on the same side of the equation
•
the x and y to the second power always have the same coefficient
•
there could be an x and/or a y to the first power (means that the center is not at the origin) as well as
•
constants
Let’s examine the equation of a circle given in Standard Form:
(
) (
)
−
2
+
+
2
=
x
2
y
3
6
9)
a) Find the center. (____, ____)
b) Find the radius. ______
c) Rewrite in General Form. _______________________________
Unit 9: Coordinate Geometry
SNRPDP
Page 1 of 3
NVACS – Revised 2014-2015
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