SOLUTION: How do I go about converting this equation to the standard form in order to graph the conic section? 4x^2-16x+9y^2+18y=0 I know I am supposed to complete the square...Thanks a bunc

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: How do I go about converting this equation to the standard form in order to graph the conic section? 4x^2-16x+9y^2+18y=0 I know I am supposed to complete the square...Thanks a bunc      Log On


   

Question 66491: How do I go about converting this equation to the standard form in order to graph the conic section? 4x^2-16x+9y^2+18y=0 I know I am supposed to complete the square...Thanks a bunch.
Answer by rapaljer(4671) About Me  (Show Source):
You can put this solution on YOUR website!
Conic Sections Summary -- ax%5E2+%2B+cy%5E2+%2Bdx+%2B+ey+%2B+f+=+0

If the coefficients of x%5E2+ and y%5E2 are of the same sign, but unequal coefficients, then the graph will be an ELLIPSE.

If both coefficients are the same sign with equal coefficients, it will be a CIRCLE.

If coefficients are of opposite sign, then it will be a HYPERBOLA.

A PARABOLA results when there is an x%5E2 but no y%5E2 or a y%5E2 but no x%5E2.

The preferred method of solving such a problem is the method of completing the square:
4x%5E2-16x%2B9y%5E2%2B18y=0

You must first factor out the coefficients of x^2 and y^2 like this:
4%28x%5E2-4x%2B____%29%2B9%28y%5E2+%2B2y+%2B+_____%29=0
4%28x%5E2-4x%2B4%29%2B9%28y%5E2+%2B2y+%2B+1%29=0%2B16%2B9
4%28x%5E2-4x%2B4%29%2B9%28y%5E2+%2B2y+%2B+1%29=25

This is an ellipse. The standard form for an ellipse is in a form = 1, so divide both sides of the equation by 25 to set it equal to 1.
%284%28x-2%29%5E2%29%2F25+%2B%289%28y%2B+1%29%5E2%29%2F25=25%2F25
%284%28x-2%29%5E2%29%2F25+%2B%289%28y%2B+1%29%5E2%29%2F25=1
Finally invert the coefficients of 4 and 9 in order to write this:
%28%28x-2%29%5E2%29%2F%2825%2F4%29+%2B%28%28y%2B+1%29%5E2%29%2F%2825%2F9%29=1

That would be standard form for an ellipse! The center is at (2,-1), with the "radius" extending 5%2F2 units in the x direction, and 5%2F3 units in the y direction.


R^2 at SCC