SOLUTION: The Equation {{{9x^2-18x+4y^2+8y=23}}} defines an ellipse with center (____, ___). The major axis has length ___ and the minor axis has length ____.

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: The Equation {{{9x^2-18x+4y^2+8y=23}}} defines an ellipse with center (____, ___). The major axis has length ___ and the minor axis has length ____.      Log On


   

Question 19417: The Equation 9x%5E2-18x%2B4y%5E2%2B8y=23 defines an ellipse with center (____, ___). The major axis has length ___ and the minor axis has length ____.
Answer by Earlsdon(6294) About Me  (Show Source):
You can put this solution on YOUR website!
You will need to get your equation into the standard form for an ellipse:
%28%28x-h%29%5E2%29%2Fa+%2B+%28%28y-k%29%5E2%29%2Fb+=+1
You can do this by completing the square in the x-terms and the y-terms. Start by groupng the x-terms together and grouping the y-terms together.
%289x%5E2+-+18x%29+%2B+%284y%5E2+%2B+8y%29+=+23 Now,factor a 9 from the x-group and factor a 4 from the y-group.
9%28x%5E2+-+2x%29+%2B+4%28y%5E2+%2B+2y%29+=+23 Complete the square in the x-terms within the parentheses by adding the square of half the x-coefficient to both sides of the equation, and do the same for the y-terms.
9%28x%5E2+-+2x+%2B+1%29+%2B+4%28y%5E2+%2B+2y+%2B+1%29+=+23+%2B+9+%2B+4
9%28x%5E2+-+2x+%2B+1%29+%2B+4%28y%5E2+%2B+2y+%2B+1%29+=+36 Now divide both sides by 36.
%28x%5E2+-+2x+%2B+1%29%2F4+%2B+%28y%5E2+%2B+2y+%2B+1%29%2F9+=+1 Factor the denominators.
%28%28x+-+1%29%5E2%29%2F4+%2B+%28%28y+%2B+1%29%5E2%29%2F9+=+1
This is the standard form of the equation for an ellipse with the center at (1, -1) and major axis is parallel to the y-axis. The length of the major axis is 3 and the length of the minor axis is 2