Geometry: An Interactive Journey to Mastery

Lesson 20: The Equation of a Circle

([DPSOH

Explain why the circle xy (^132) ©¹ ̈ ̧§· 2125242 must be tangent to the x-axis.
6ROXWLRQ
The center of the circle is ̈ ̧§·©¹13, 2^12 , and its radius rVDWLV¿HVr^2 254 , giving r 5122 2.
The center of the circle is the same distance above the x-axis as the radius of the circle. The circle must
therefore be tangent to the axis.
6WXG\7LS
x $TXLFNVNHWFKRIDULJKWWULDQJOHVKRZVWKDWIRUͼ௘x, y௘ͽWREHDGLVWDQFHrIURPDFHQWHUSRLQWͼ௘a, b௘ͽZH
KDYHͼ௘xía௘ͽ^2 ͼ௘yíb௘ͽ^2 = r^2. There is no need to memorize the equation of a circle. Just see it as yet
another application of the Pythagorean theorem.
3LWIDOO
x 7KHHTXDWLRQͼ௘x௘ͽ^2 ͼ௘y௘ͽ^2 IRUH[DPSOHUHSUHVHQWVDFLUFOHZLWKFHQWHUͼ௘íí௘ͽDQG
radius r 28. Watch out for two things: We need the difference of coordinates on the left part of the
equation and the radius squaredRQWKHULJKWSDUWRIWKHHTXDWLRQͼ௘7KHVHREVHUYDWLRQVPDNHVHQVHZLWK
DVNHWFKDQGWKH3\WKDJRUHDQWKHRUHP௘ͽ
 Write the centers and radii of the following circles.
D௘ͽ xy 2223 16.
E௘ͽ xy 755.^22
F௘ͽ xy 7221 10.
G௘ͽ xy 252 2 1.
H௘ͽ xy^22  19.
I௘ͽ xy 172217 17.
Problems