Geometry: An Interactive Journey to Mastery

The Equation of a Circle

Lesson 20

Topics
x The equation of a circle.
x Pythagorean triples.
)RUPXOD
x The equation of a circle with center C ͼ௘a, b௘ͽDQGUDGLXVrLVͼ௘xía௘ͽ^2 ͼ௘yíb௘ͽ^2 = r^2 ͼ௘7KLVLVMXVWDQ
DSSOLFDWLRQRIWKH3\WKDJRUHDQWKHRUHP௘ͽ
Summary
,QWKLVOHVVRQZH¿QGDQHTXDWLRQWKDWPXVWEHWUXHIRUDSRLQWͼ௘x, y௘ͽWROLHRQDFLUFOHZLWKDJLYHQFHQWHUDQGD
given radius. As a bonus, we explore an interesting connection to Pythagorean triples.
([DPSOH
:KDWLVWKHHTXDWLRQRIDFLUFOHZLWKFHQWHUͼ௘௘ͽDQGUDGLXV",VWKHSRLQWͼ௘௘ͽRQWKHFLUFOH",VWKHSRLQW
ͼ௘í௘ͽRQWKHFLUFOH"
6ROXWLRQ
The equation is x^2 ͼ௘yí௘ͽ^2 = 29^2 = 841. Noting that 20^2 ͼ௘í௘ͽ^2 = 20^2 + 21^2 = 400 + 441 = 841, we see that
ͼ௘௘ͽGRHVLQGHHG¿WWKHHTXDWLRQVRWKLVSRLQWOLHVRQWKHFLUFOH&RPSXWLQJͼ௘í௘ͽ^2 + 7^2 = 149  841 shows
WKDWWKHSRLQWͼ௘í௘ͽLVQRWRQWKHFLUFOHͼ௘,QIDFWLWLVLQVLGHWKHFLUFOH௘ͽ
([DPSOH
)LQGWKHHTXDWLRQRIWKHFLUFOHZLWKFHQWHUͼ௘ʌíʌ௘ͽWKDWSDVVHVWKURXJKWKHSRLQWͼ௘ʌ௘ͽ
6ROXWLRQ
The radius of the circle is the distance between the two given points—namely, SS S^22  2.^2 Thus, the
equation of the circle is xy SSS^2 2 2.^22 That is,
xy SSS^22 2.^2