Conic Equations

Question

I'm confused as to how you identify which equation for a conic is being used. For example, an ellipse has two equations, $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2}=1$ or $\frac{(y-k)^2}{a^2} + \frac{(x-h)^2}{b^2} =1$ and this determines which direction the major axis goes. However, I'm very confused as to how you would tell the difference between the two. When $b^2$ and $a^2$ are replaced with numbers, how do you identify between the two equations?

Answer 1

Those two equations are effectively identical, in that each becomes the other when $a$ and $b$ are exchanged. That is, the distinction is only meaningful if you have a "hard-wired" correspondence between geometric concepts and symbols.

For example, many people use $(x, y)$ to denote Cartesian coordinates in the plane, with $x$ measuring horizontal position, increasing to the right, and $y$ measuring vertical position, increasing upward.

Perhaps you're similarly assuming $0 < a < b$, so that the major axis is always called "$b$" (or vice versa)...? If that's what you mean, take $b$ to be the square root of the larger denominator. For example, if $$ \frac{x^{2}}{5} + 7y^{2} = \frac{x^{2}}{(\sqrt{5})^{2}} + \frac{y^{2}}{(1/\sqrt{7})^{2}} = 1, $$ then $a = 1/\sqrt{7}$ and $b = \sqrt{5}$, since $1/\sqrt{7} < \sqrt{5}$.