ellipse polar co-ordinate conversion I have a somewhat trivial question out of interest. Given the equation of an ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ why is the substitution $x = \sqrt{a}\cos t$ and $y = \sqrt{b}\sin t$ valid? For the unit circle $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ it is clear graphically why the polar co-ordinate conversion $x = r \cos t$ and $y = r \sin t$ is valid since the radius $r$ is fixed, but with an ellipse, this is not the case. Why is it taken trivially that this substitution is valid? Thanks for any assistance.

You don't actually have these square roots there. The line of thought is like this: $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \iff \left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = 1,$$so making a change of coordinates $\overline{x} = x/a$ and $\overline{y} = y/b$, we have that this equation reads $\overline{x}^2+\overline{y}^2 = 1$. That ellipse in the $xy$ plane is a circle in the $\overline{x}\overline{y}$ plane, with the axes stretched. We can parametrize it as ${\overline{x}} = \cos t$, $\overline{y}= \sin t$. Going back to the $xy$ coordinates we obtain $x = a \cos t$ and $y = b \sin t$.