1. Ellipse Equation

The graph of which equation forms an ellipse?
(1) x^2 - y^2 = 9 (3) 2x^2 + y^2 = 8
(2) 2x^2 + 2y^2 = 8 (4) xy = -8

I tested myself on a textbook quiz and got this question wrong. I selected choice (2) but the correct answer is choice (3). What makes choice (3) correct?

2. An ellipse has the equation: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.

You can see that for equation 2, dividing by 2 gives $x^2 + y^2 = 4$. Because the coefficients of $x^2$ and $y^2$ are the same, this equation represents a circle.

For equation 3, dividing by 8 gives $\frac{x^2}{4} + \frac{y^2}{8} = 1$
The coefficients of $x^2$ and $y^2$ are different, and so this equation represents an ellipse

3. Yes............

Originally Posted by nzmathman
An ellipse has the equation: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.

You can see that for equation 2, dividing by 2 gives $x^2 + y^2 = 4$. Because the coefficients of $x^2$ and $y^2$ are the same, this equation represents a circle.

For equation 3, dividing by 8 gives $\frac{x^2}{4} + \frac{y^2}{8} = 1$
The coefficients of $x^2$ and $y^2$ are different, and so this equation represents an ellipse
Yes, I can clearly see that now thanks to you.

4. Technically, a circle is an ellipse - just a "special" case of one where the two foci happen to be at the same point (or in terms of nzmathman's equation, the "special" case where a = b)

5. yes...

Originally Posted by o_O
Technically, a circle is an ellipse - just a "special" case of one where the two foci happen to be at the same point (or in terms of nzmathman's equation, the "special" case where a = b)
Yes, I read that a circle is a special type of circle in the textbook some months ago.