Unformatted text preview: in the
plane whose distance to (h, k ) is r. (x, y )
r
(h, k ) From the picture, we see that a point (x, y ) is on the circle if and only if its distance to (h, k ) is r.
We express this relationship algebraically using the Distance Formula, Equation 1.1, as
r= (x − h)2 + (y − k )2 By squaring both sides of this equation, we get an equivalent equation (since r > 0) which gives us
the standard equation of a circle.
Equation 7.1. The Standard Equation of a Circle: The equation of a circle with center
(h, k ) and radius r > 0 is (x − h)2 + (y − k )2 = r2 .
Example 7.2.1. Write the standard equation of the circle with center (−2, 3) and radius 5.
Solution. Here, (h, k ) = (−2, 3) and r = 5, so we get
(x − (−2))2 + (y − 3)2 = (5)2
(x + 2)2 + (y − 3)2 = 25 Example 7.2.2. Graph (x + 2)2 + (y − 1)2 = 4. Find the center and radius.
Solution. From the standard form of a circle, Equation 7.1, we have that x + 2 is x − h, so h = −2
and y − 1 is y − k so k = 1. This tells us that our center is (−2, 1). Furthermore, r2 = 4, so r = 2.
Thus we have a circle centered at (−2, 1) with a radius of 2. Graphing gives us 7.2 Circles 401
y
4
3
2
1
−4 −3 −2 −1
−1 1 x If we were to expand the equation in the previous example and gather up like terms, instead of the
easily recognizable (x + 2)2 + (y − 1)2 = 4, we’d be contending with x2 + 4x + y...
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 Fall '13
 Wong
 Algebra, Trigonometry, Cartesian Coordinate System, The Land, The Waves, René Descartes, Euclidean geometry

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