Notes
A circle is the set of points at distance $r$ from a center $(h, k)$.
- Standard form: $(x - h)^2 + (y - k)^2 = r^2$.
- To convert from general form, group x terms and y terms and complete the square.
- Example: $x^2 + y^2 - 6x + 4y - 3 = 0$ becomes $(x - 3)^2 + (y + 2)^2 = 16$, so the center is $(3, -2)$ and the radius is 4.
Practice
What are the center and radius of $(x + 1)^2 + (y - 4)^2 = 25$?
The signs flip inside the parentheses, and $r^2 = 25$ gives $r = 5$.
Which is the equation of the circle with center $(2, -3)$ and radius 6?
Use $(x - h)^2 + (y - k)^2 = r^2$ with $h = 2$, $k = -3$, $r^2 = 36$.
What is the radius of $x^2 + y^2 - 6x + 4y - 3 = 0$?
Complete the square: $(x - 3)^2 + (y + 2)^2 = 3 + 9 + 4 = 16$.
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