I am trying to find the standard form of this equation for an ellipse.
9x^2 + 4y^2 - 36x + 8y + 31 = 0
First I grouped the x and y's together.
9x^2 - 36x + ? + 4y^2 + 8y + ? = -31 + ? + ?
Then I factored the 9 and 4.
9(x^2-4x+4) + 4(x^2+2x+1) = -31+36+4
9(x-2)^2 + 4(x-1)^2 = 9
Then I divided it all by 9 and got:
(x-2)^2 / 1 + 4(x-1)^2 / 9 = 1
Is this correct or did I mess up somewhere?
Your steps are mostly correct, but there is a small mistake. Let's go through the steps to find the standard form of the equation for an ellipse and correct the mistake.
Step 1: Group the x and y terms together:
9x^2 - 36x + 4y^2 + 8y = -31
Step 2: Complete the square for both the x and y terms.
To complete the square for the x terms, take half of the coefficient of x (-36/2 = -18), square it (-18^2 = 324), and add it to both sides of the equation.
To complete the square for the y terms, take half of the coefficient of y (8/2 = 4), square it (4^2 = 16), and add it to both sides of the equation.
9(x^2 - 4x + 4) + 4(y^2 + 2y + 1) = -31 + 324 + 16
Step 3: Simplify the equation:
9(x - 2)^2 + 4(y + 1)^2 = 309
Step 4: Divide both sides of the equation by the constant on the right side to make it equal to 1:
9(x - 2)^2/309 + 4(y + 1)^2/309 = 1
So, the corrected standard form of the ellipse equation is:
(x - 2)^2/34.333 + (y + 1)^2/77.25 = 1
Note: In the corrected equation, the constant on the right side is used to normalize the equation to 1. Also, keep in mind that the standard form of an ellipse equation is generally written as (x - h)^2/a^2 + (y - k)^2/b^2 = 1, where (h, k) represents the center of the ellipse and 'a' and 'b' represents the semi-major and semi-minor axes lengths, respectively.