The standard form equation is: (x + 1)² / 9 + (y - 1/3)² / 16 = 1.
Here's how to find the standard form equation for the ellipse:
1. Complete the square for the x² term:
Move the constant term to the right side: 2x² + 3y² = 4 + 4x - 12y
Divide both sides by 2 to isolate x²: x² = 2 + 2x - 6y
Add (2/2)² to both sides to complete the square: x² + (1)² = 4 + 2x - 6y + (1)²
Rewrite as (x + 1)² = 9
2. Complete the square for the y² term:
Divide both sides by 3 to isolate y²: y² = 4/3 + 4x/3 - 2y/3
Multiply both sides by 3 to get rid of fractions: 3y² = 12 + 4x - 2y
Add (-2/2)² to both sides to complete the square: 3y² + (1)² = 13 + 4x - 2y + (1)²
* Rewrite as 3(y - 1/3)² = 16
3. Combine the completed squares:
Divide both sides by 9 to get the equation in standard form: (x + 1)²/9 + (y - 1/3)²/16 = 1
Therefore, the standard form equation for the ellipse is:
(x + 1)² / 9 + (y - 1/3)² / 16 = 1