Convert these from general form to standard form.

Question

Determine the standard form of equation of the ellipse whose general form is given.

1. 4x² + 3y² - 8x +6y - 5 = 0
2. 9x² + 10y² + 54x + 100y +241 = 0
3. 4x² + 9x² - 32x - 36y +64 = 0
4. 5x² + 9y² +20x -180y + 830 = 0
5. 64x² + 9y² - 128x - 36y - 44 = 0

Answer 1

Steps to solve including problem 1 as an example:

We want to put the equation into the standard for of (x-h)² / a² + (y-k)² / b² = 1, where (h,k) is the center of the ellipse and a/b are the horizontal/vertical radii for the ellipse.

1. Group x terms, y terms, and move number term to the other side.
2. (4x² - 8x) + (3y² +6y) = 5
3. Factor out the coefficients for x² and y²
4. 4(x² - 2x) + 3(y² + 2y) = 5
5. Complete the square within the x and y parentheses, using (b/2)²
6. For (x² - 2x), (b/2)² = 1, which gives us (x² - 2x +1). Now we factor using perfect squares to get (x-1)²
7. In case that's confusing, let's multiply out (x-1)² to make sure it works: (x-1)(x-1) = x² -x -x +1 = x² -2x +1.
8. For (y² + 2y), (b/2)² = 1 as well. We factor (y² + 2y +1) using perfect squares to get (y+1)².
9. Balance the equation by adding everything we added on the left side to the right side. Note: the numbers we add to the left side are technically factored already, so make sure to multiply them.
10. We added 1 to 4(x² - 2x), so the 1 needs to be multiplied by 4 before being added to the other side. We added 1 to 3(y² + 2y), so it becomes a 3. Overall we must add 7 to the right side.
11. We now have the equation 4(x² - 2x) + 3(y² + 2y) = 12.
12. The right side needs to equal 1, so we divide by the number we have there now.
13. Everything is divided by 12, yielding (x² - 2x) / 3 + (x² - 2x) / 4 = 1. We are done.