SOLUTION: 9x^2+4y^2+36x-24y+36=0

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: 9x^2+4y^2+36x-24y+36=0       Log On


   

Question 1197383: 9x^2+4y^2+36x-24y+36=0
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
9x%5E2%2B4y%5E2%2B36x-24y%2B36=0
%289x%5E2%2B36x%29%2B%284y%5E2-24y%29%2B36=0.............complete squares
9%28x%5E2%2B4x%29%2B4%28y%5E2-6y%29%2B36=0
9%28x%5E2%2B4x%2Bb%5E2%29-9b%5E2%2B4%28y%5E2-6y%2Bb%5E2%29-4b%5E2%2B36=0
9%28x%5E2%2B4x%2B2%5E2%29-9%2A2%5E2%2B4%28y%5E2-6y%2B3%5E2%29-4%2A3%5E2%2B36=0
9%28x%2B2%29%5E2-36%2B4%28y-3%29%5E2-36%2B36=0
9%28x%2B2%29%5E2%2B4%28y-3%29%5E2-36=0
9%28x%2B2%29%5E2%2B4%28y-3%29%5E2=36............divide by 36
9%28x%2B2%29%5E2%2F36%2B4%28y-3%29%5E2%2F36=1
%28x%2B2%29%5E2%2F4%2B%28y-3%29%5E2%2F9=1
=> it’s an vertical ellipse with center at (-2,3)
=> a%5E2=9 and b%5E2=4
=> a=3 and b=2
semi-major axis length: 3
semi-minor axis length: 2






Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The same solution as shown by the other tutor... but I find her presentation of the process a bit messy.

9x%5E2%2B4y%5E2%2B36x-24y%2B36=0

Move the constant to the other side and complete the square in both x and y.

Be sure when completing the square that you add the same constants to both sides of the equation.

%289x%5E2%2B36x%29%2B%284y%5E2-24y%29=-36
9%28x%5E2%2B4x%29%2B4%28y%5E2-6y%29=-36
9%28x%5E2%2B4x%2B4%29%2B4%28y%5E2-6y%2B9%29=-36%2B9%284%29%2B4%289%29=-36%2B36%2B36
9%28x%2B2%29%5E2%2B4%28y-3%29%5E2=36

Put in standard form by dividing by the constant on the right, to make the right side equal to 1.

%28x%2B2%29%5E2%2F2%5E2%2B%28y-3%29%5E2%2F3%5E2=1

That is the standard form of an equation with center (-2,3), vertical semi-major axis of length 3, and horizontal semi-minor axis of length 2.