Ellipse Equations?

Question

Explain how you can tell by inspection if an equation in GENERAL FORM (ex: 4x² + 9y² - 8x - 54y + 49 = 0) is a circle or an ellipse, as well explain how you can tell by inspection of an equation in general form has its centre at the origin.

Answer 1

To be a circle, the coefficients on the x^2 and y^2 terms must be the same and non-zero. If the coefficients differ, but have the same sign (both positive or both negative), it is an ellipse.

To be centered on the origin, in addition to meeting the requirements above, there must be no x or y term. (That's because a circle centered at (h,k) is (x-h)^2 + (y-k)^2= r^2 -- and if h and k are zero there is no term with x and y.)

Using those rules, your sample equation is
(a) an ellipse (x^2 and y^2 coefficients are 4 and 9 -- same sign and non-zero, but not the same number)
(b) NOT centered on the origin (x and y terms are present, with coefficients -8 and -54).

Note: I assume "by inspection" means that you aren't supposed to be doing any significant operations on the equations (e.g., completing squares, factoring, etc.).

Answer 2

Complete the square for both variables. The equation of an ellipse looks like this: (x-h)^2/a^2 + (y-k)^2/b^2 = 1, where a is the major axis and b the minor. The major axis is parallel to the x or horizontal axis in this case. If it looked like this: (x-h)^2/b^2 + (y-k)^2/a^2 = 1, the major axis would be parallel to the y or vertical axis. The center is at (h,k).

So we have, 4x² + 9y² - 8x - 54y + 49 = 0.

Complete the square for both variables:
4(x^2 -2x+1) + 9(y^2-6y+9) = -49 + 4 + 81

4(x-1)^2+9(y-3)^2=36

Which gives you: (x-1)^2/9+(y-3)^2/4 = 1

Center is at (1,3) and a= 3 and b = 4.
c^2=a^2-b^2, so that would give you c = sqrt(5), in case you need that.

Answer 3

you can find out by completing the squares of the terms:

4x² + 9y² - 8x - 54y + 49 = 0
4x^2 - 8x + 9y^2 -54y +49 = 0
4 ( x^2 - 2x ) + 9 ( y^2 - 6y ) + 49 = 0
4 ( x^2 - 2x + 1 ) + 9 (y^2 - 6y + 9 ) + 49 - 4 - 81 = 0
4 (x - 1)^2 + 9 (y - 3)^2 = 36
(x -1)^2 / 9 + (y-3)^2 / 4 = 1

so this is an ecllipse, not a circle!