Integrated Math 2 / Unit 8: Circles
8.6 Equation of a Circle
Notes
A circle with center $(h, k)$ and radius $r$ has equation $(x - h)^2 + (y - k)^2 = r^2$.
- Center $(2, -3)$ and radius 4: $(x - 2)^2 + (y + 3)^2 = 16$. Note the sign flip and that $r$ is squared.
- Center at the origin: $x^2 + y^2 = r^2$.
- To read a circle from $x^2 + y^2 - 6x + 4y - 12 = 0$, complete the square: $(x - 3)^2 + (y + 2)^2 = 25$. Center $(3, -2)$, radius 5.
Practice
Write the equation of the circle with center $(2, -3)$ and radius 4.
Subtract the center coordinates: $x - 2$ and $y - (-3)$. The right side is $r^2 = 16$.
What are the center and radius of $(x + 1)^2 + (y - 5)^2 = 49$?
Flip the signs for the center, and $\displaystyle r = \sqrt{49} = 7$.
Find the center and radius of $x^2 + y^2 - 6x + 4y - 12 = 0$.
Complete the square: $(x - 3)^2 + (y + 2)^2 = 12 + 9 + 4 = 25$.
Ready for more? Take the Unit 8 test.