Equations of Circles

Math is fun, right? If you know everything. Here is a simple overview about writing and solving equations for circles.

In the figure to the left, there is a circle shown with center (4,6).

In its standard form, a circle's equation is:
(x-h)² + (y-k)² = r², with (h,k) being the center of the circle and r being the radius.

The circle to the left has r=2, h=4 and k=6, as the center is (4,6). So, the factored form of the equation would be (x-4)² + (y-6)² = 4

Now, another problem you might have is when the equation isn't factored, such as:

x² + y² - 14x + 8y + 16

In this case, you need to complete the square to make the equation into standard form. To complete the square, first pair up the x's together and the y'x together. In the above equation, this means doing:

(x² - 14x) + (y² + 8y) = -16

Then, the coefficient behind the variables with degree 1 need to be halved then squared to get the constant that creates a quadratic equation. In this case:

(x² - 14x + 7²) + (y² + 8y + 4²) = 16 + 49 + 16

which, simplified, is:

(x² - 14x + 49) + (y² + 8y + 16) = 81

Then, just factor out the quadratic equations to get:

(x-7)² + (y+4)² = 81

Obviously, this is the standard form encountered earlier. Therefore, this circle's center is (7, -4), and the radius is sq. root of 81, or 9.