Conic equation

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 !!!

Lets take a look.

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

$$9x^2+36x+4y^2-24y=-36$$

$$9(x^2+4x)+4(y^2-6y)=-36$$

$$9(x^2+4x+4)+4(y^2-6y+9)=-36+(9*4)+(4*9)\\

9(x+2)^2+4(y-3)^2=-36+36+36\\

9(x+2)^2+4(y-3)^2=36\\

\mbox{Divide throughout by 36}\\$$

$$\frac{x+2}{4}+\frac{y-3}{9}=1$$

That looks like an ellipse to me - centred at (-2,3)

this is what it looks like

https://www.wolframalpha.com/input/?i=%283%28x%2B2%29%29%5E2%2B%282%28y-3%29%29%5E2%3D36

Awe - Don't I get a tick too?  

Everyone's "ticked off" now !!!

Thanky you Chris