What is major axis for the equation 9x^2 + 16y^2 - 36x + 96y + 36 = 0
To determine the major axis of the equation 9x^2 + 16y^2 - 36x + 96y + 36 = 0, we need to rewrite it in the standard form of an ellipse equation.
Step 1: Group the x-terms and y-terms separately:
9x^2 - 36x + 16y^2 + 96y = -36
Step 2: Complete the square for both the x-terms and y-terms. Starting with the x-terms:
9(x^2 - 4x) + 16y^2 + 96y = -36
Divide the coefficient of x by 2, square it, and add/subtract it inside the parentheses:
9(x^2 - 4x + 4) + 16y^2 + 96y = -36 + 36
Do the same for the y-terms:
9(x^2 - 4x + 4) + 16(y^2 + 6y) = 0
Now simplify:
9(x - 2)^2 + 16(y + 3)^2 = 0
Step 3: Divide both sides of the equation by the constant term (36) to isolate the ellipse:
(x - 2)^2/[(0.49)^2] + (y + 3)^2/[(0.42)^2] = 1
Now, the equation can be written as:
[(x - 2)^2]/[(0.49)^2] + [(y + 3)^2]/[(0.42)^2] = 1
Comparing the equation to the standard form [(x - h)^2/a^2] + [(y - k)^2/b^2] = 1, we can see that the major axis is along the x-axis.
The length of the major axis is equal to 2a, where a represents the length of the semi-major axis. In this case, a is equal to 0.49.
Therefore, the length of the major axis is 2 * 0.49 = 0.98.
Hence, the major axis for the equation 9x^2 + 16y^2 - 36x + 96y + 36 = 0 is 0.98.
I will complete the square for you for this ellipse, you should be able to figure it from there
9(x^2 -4x + ...) + 16(y^2 + 6y + ...) = -36
9(x^2 - 4x + 4) + 16(y^2 + 6y + 9) = -36 + 36 + 144
9(x-2)^2 + 16(y+3)^2 = 144
(x-2)^2/16 + (y+3)^2/9 = 1