Completing the Ellipse …I mean square. In an ellipse.

So what’s new to this process? Compared to circles, we just need to worry about two different steps. First, near the beginning, we need to factor anything off of the x2 and y2 terms. Second, at the end, we have to divide by whatever’s on the other side from the variables (so that it equals 1).

Example 1 4x2 + 9y2 = 36 Divide both sides by 36 x2 + y2 = 1 9 4

Example 2: Steps 1 and 2 4x2 + 9y2 + 16x – 18y – 11 = 0 First thing: Collect like terms and group (4x2 + 16x) + (9y2 – 18y) – 11 = 0 Move the – 11 over – how? Add 11 to both sides (4x2 + 16x) + (9y2 – 18y) = 11

Example 2: Step 3 (4x2 + 16x) + (9y2 – 18y) = 11 What next? Factor! Whatever’s on the x2 and the y2 out of their parentheses. How do we do that again? Divide everything inside the parentheses by the number, which then goes on the outside. 4(x2 + 4x) + 9(y2 – 2y) = 11

Example 2: Step 4 4(x2 + 4x) + 9(y2 – 2y) = 11 Now what? Complete the square! How? (-b/2a)2 Add that number inside the parentheses. (-(4)/2(1)2 = -22 = 4 (-(-2)/2(1))2 = 12 = 1 4(x2 + 4x + 4) + 9(y2 – 2y + 1) = 11

Example 2: Step 4.5 4(x2 + 4x + 4) + 9(y2 – 2y + 1) = 11 I just added something to one side. What else do I need to do? Add it to the other side! But, wait: I also need to compensate for the numbers in front of the parenthesis. Multiply when adding to the other side. 4(x2 + 4x + 4) + 9(y2 – 2y + 1) = 11+(4)(4)+(9)(1)

Example 2: Step 5 4(x2 + 4x + 4) + 9(y2 – 2y + 1) = 36 I added the constants. Now what? Simplify the stuff in parentheses – should become (x – h)2 and (y – k)2, where h and k are the numbers we squared to get the 4 and 1. 4(x – (-2))2 + 9(y – (1))2 = 36 4(x + 2)2 + 9(y – 1)2 = 36

Example 2: Step 6 (last step) 4(x + 2)2 + 9(y – 1)2 = 36 Last step? Need it to equal 1. So divide by 36 And….boo-ya.