Completing the Square: Circle Equations


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Completing the Square: Circle Equations

The technique of completing the square is used to turn a quadratic into a squared binomial, plus some loose numbers: (x – a)² + b. Since the center-radius form of the circle equation is in the format (x – h)² + (y – k)² = r² (with the center being at the point (h, k) and the radius being "r"), and since the circle equation is often given in the so-called "standard" format of ax² + by² + cx + dy + e = 0, completing the square can be necessary for finding the center and radius of a circle, once you've been given the equation. Here's how it works. Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

Find the center and radius of the circle having the following equation:
4x² + 4y² – 16x – 24y + 51 = 0.

Here is the equation they've given you.
Move the loose number over to the other side.
Group the x-stuff together. Group the y-stuff together.
Whatever is multiplied on the squared terms (it'll always be the same number!), divide it off from every term.
This is the complicated step. You'll need space inside your groupings, because this is where you'll add the squaring term. Take the x-term coefficient, multiply it by one-half, square it, and then add this to both sides of the equation, as shown. Do the same with the y-term coefficient. Convert the left side to squared form, and simplify the right side.
Read off the answer from the rearranged equation.	The center is at (h, k) = (x, y) = (2, 3). The radius is r = sqrt(¹/₄ ) = ¹/₂

Completing the square to find a circle's center and radius always works in this manner. Always do the steps in this order, and each of your exercises should work out fine. (Also, if you get in the habit of always working the exercises in the same manner, you are more likely to remember the procedure on tests.)

About the only fiddly parts are in the interpretation of the final equation. Remember that the circle formula is (x – h)² + (y – k)² = r², so if you have something like (x + 4)² + (y + 5)² = 5, you have to keep straight that h and k are subtracted, so you really have (x – (–4))² + (y – (–5))² = 5. That is, the center is at the point (–4, –5), not at (4, 5). Be careful with the signs; don't just "read off the answer" without thinking. Also, remember that the formula says "r²", not "r", so the radius in this case is sqrt(5), not 5.

In the course of the above procedure, about the only other thing that can be a problem is forgetting the sign on the step where you multiply by one-half. If you drop a negative, you'll get the wrong answer for the coordinates of the center, so be careful of this. Don't try to do this step in your head: write it out!

Here's one more example of how completing the square works for circle equations:

Find the center and radius of the circle with the following equation:
100x² + 100y² – 100x + 240y – 56 = 0.

This is the given equation.
Move the loose number over to the other side.
Group the x-stuff and y-stuff together.
Divide off by whatever is multiplied on the squared terms.
Take the coefficient on the x-term, multiply by one-half, square, and add inside the x-stuff and also to the other side. Do the same with the y-term.
Convert the left-hand side to squared form, and simplify the right-hand side.
If necessary, fiddle with signs and exponents to make your equation match the circle equation's format.
Read off the answer.	The center is at ( ¹/₂, –⁶/₅ ) and the radius is ³/₂.