Decentered Circles – Determine General Form by Pattern Matching
Prof. Townsend
Fall 2015
The general form of a decentered circle is, as given in section 21.3 of Washington,
(x − h)2 + (y − k)2 = r 2
(1)
The problem is to figure out h, k, and r from a problem presented in the form
Ax 2 + Ay 2 + Dx + Ey + F = 0
(2)
Using equation 6.4 in Washington we expand then rearrange equation (1).
x 2 + y 2 − 2hx − 2ky + h 2 + k 2 = r 2
(3)
The goal is compare equations (2) and (3) to find h, k, and r from pattern matching.
First move the r 2 to the left to get
x 2 + y 2 − 2hx − 2ky + h 2 + k 2 − r 2 = 0
(4)
Recall equation (2)
Ax 2 + Ay 2 + Dx + Ey + F = 0
(2)
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Note that in equation (2) both x 2 and y 2 have an A in front of them while in equation (4) there is
nothing in front of them. So, divide equation (2) by A.
Ax 2 + Ay 2 + Dx + Ey + F 0
=
A
A
simplify
x 2 + y2 +
D
E
F
x+ y+ = 0
A
A
A
(5)
To simplify the notation, we rewrite equation (5) as
x 2 + y 2 + Dx + Ey + F = 0
(6)
where D, E, and F now give us the adjusted general form for the decentered circle.
Compare equation (6) with equation (4)
x 2 + y 2 + Dx + Ey
+
F
=0
(6)
x 2 + y 2 − 2hx − 2ky + ( h 2 + k 2 − r 2 ) = 0
(4)
Match coefficients of x and y and identify F. By inspection we have
D = −2h
F = h2 + k 2 − r 2
E = −2k
(7)
First solve for h and k. Then we can solve for r.
h=−
D
2
k=−
E
2
r 2 = h2 + k 2 − F
Now we solve some problems
Example 5 page 577:
x 2 + y 2 − 6x + 8y − 24 = 0
Line up with equation (6) to get D, E, and F.
D = −6
E=8
F = −24
−6
=3
2
8
k = − = −4
2
h=−
r 2 = ( 3) + ( −4 ) − ( −24 ) = 49
r=7
2
Page 2 of 3
2
(8)
Plug these numbers into equation (1)
(x − h)2 + (y − k)2 = r 2
(1)
( x − 3)2 + ( y − ( −4 )) = 7 2
( x − 3)2 + ( y + 4 )2 = 7 2
2
or
(h, k) = (3,−4) with radius r = 7
The center of the circle is at
3x 2 + 3y 2 − 9.60y − 2.80 = 0
Example 6 page 577:
First we need to get rid of the coefficients of x 2 and y 2 . Dividing through by 3 we get
x 2 + y 2 − 3.20y − 0.933 = 0
This can be rewritten as
x 2 + y 2 + 0.00x − 3.20y − 0.933 = 0
Line up with equation (6) to get D, E, and F.
D = 0 since there is no x term.
E = 3.20
F = −0.933
0
=0
2
−3.20
k=−
= 1.60
2
h=−
r 2 = ( 0.00 ) + (1.60 ) − ( −0.933) = 3.493
2
2
r = 3.493 = 1.87
Plug these numbers into equation (1)
(x − h)2 + (y − k)2 = r 2
( x ) + ( y − 1.60 )
2
2
(1)
= 3.493
The center of the circle is at
(h, k) = (0.00,1.60)
Page 3 of 3
with radius r = 1.87