When doing a problem where you have to find the foci and vertices of an ellipse given and equation like $9x^2-36x+4y^2=0$, The answer will change based on if they are or not, so how do you tell from that if the foci are located on the y axis or not?


Foci lie on the long axis.

To find where the long axis is, rewrite the equation by completing the square:

$$\frac{1}{4}(x - 2)^2 + \frac{1}{9}y^2 = 1$$

The long axis (where the foci are) is $x = 2$, which is not $y$ axis.


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