10.7 Equation of a Circle
Notes
A circle with center $(h, k)$ and radius $r$ has the equation below. It comes from the distance formula.
$$(x - h)^2 + (y - k)^2 = r^2$$
- Watch the signs: $(x + 3)^2$ means $h = -3$.
- To find the center and radius from an expanded equation, complete the square for $x$ and for $y$.
- Example: $x^2 + y^2 - 6x + 4y = 12$ becomes $(x - 3)^2 + (y + 2)^2 = 25$: center $(3, -2)$, radius 5.
Practice
What are the center and radius of $(x + 4)^2 + (y - 1)^2 = 49$?
$h = -4$, $k = 1$ and $\displaystyle r = \sqrt{49} = 7$.
Write the equation of the circle with center $(2, -5)$ and radius 3.
Use $(x - h)^2 + (y - k)^2 = r^2$ with $r^2 = 9$.
Find the radius of $x^2 + y^2 + 8x - 2y = 8$.
$(x + 4)^2 + (y - 1)^2 = 8 + 16 + 1 = 25$, so $r = 5$.
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