9x^2+4y^2+36x-24y+36=0
OK...hang on !!
Let's rewrite this as
9x^2 + 36x + 4y^2 -24y = -36 Now factoring out the "9" and the "4" we have
9(x^2 + 4x + ) + 4(y^2 - 6y + ) = -36 (A)
Now, take half of "4", square it and add the result (4) into the first parentheses
9(x^2 + 4x + 4 ) + 4(y^2 - 6y + ) = -36
And take half of the -6 in the second parentheses, square it and add the result (9) into the second parentheses
9(x^2 + 4x + 4 ) + 4(y^2 - 6y + 9 ) = -36
Notice, on the left side we've actually added 9(4) and 4(9) = 72 to the original expression (A)
So, I need to add this to the right side, too. And we get
9(x^2 + 4x + 4 ) + 4(y^2 - 6y + 9 ) = -36 + 72 And we have
9(x^2 + 4x + 4 ) + 4(y^2 - 6y + 9 ) = 36
Now divide everything on both sides by 36 This gives
(x^2 + 4x + 4 ) / 4 + (y^2 - 6y + 9) / 9 = 1
Now, factor the stuff in both sets of parentheses ...... We get ......
(x + 2)^2 / 4 + (y - 3)^2 / 9 = 1
And we have an ellipse centered at (-2, 3) with a minor (horizonta)l axis of 4 and a major (vertical) axis of 6.
And that's it !!!
