1. ## Calculation of y^n.

I'm doing some exercises and I came across this one that starts requesting an implicit differentiation of the second derivative:

$\displaystyle x^2a^2+y^2b^2=1$

I've done a few of these implicit differentiation by now but what's confusing me about this one is that the first step of the solutions manual transforms the equation into:

$\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Later of course the right side of the equation is set to 0 and x' and y' are calculated. This first step above though has me wondering how the first equation which is a product of x^2a^2 and y^2b^2 gets changed to a division of each respectively looks strange to me. I'm wondering if anyone may have a better understanding of how those might be equivalent?

It looks like the solution's manual was suggesting that so long as $\displaystyle a$ and $\displaystyle b$ are nonzero, this is the equation for an ellipse. If $\displaystyle a\neq 0$ then $\displaystyle \dfrac{1}{a}\neq 0$. So, the solutions manual really should have written $\displaystyle \dfrac{x^2}{a_1^2} + \dfrac{y^2}{b_1^2}=1$ where $\displaystyle a_1 = \dfrac{1}{a}$ and $\displaystyle b_1 = \dfrac{1}{b}$. In other words, they are a different $\displaystyle a$ and $\displaystyle b$.