Mapping Circle to Ellipse with Dilation?

[Stefk]

Homework Statement

The problem comes from S. Lang's "Basic mathematics", chapter 7, §1:

"Consider the following generalization of a dilation. Let $a > 0, b > 0$. To each point $(x, y)$ of the plane, associate the point $(ax, by)$. Thus we stretch the x-coordinate by $a$ and the y-coordinate by $b$. This association is a mapping which we may denote by $F_{a,b}$.

Show that the set of points $(u, v)$ satisfying the equation $$\left( \frac u a \right)^2 + \left( \frac v b \right)^2 = 1$$ is the image of the circle of radius 1 centered at $O$ under the map $F_{a,b}$."

Homework Equations

Equation of a circle of radius $r$ centered at $(a, b)$: $$(x - a)^2 + (y - b)^2 = r^2$$
Equation of a circle of radius $r$ centered at the origin: $$x^2 + y^2 = r^2$$
Equation of a circle of radius 1 centered at the origin: $$x^2 + y^2 = 1$$

The Attempt at a Solution

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I've spent quite some time on this problem and I still can't see how to solve it. I understand by subsequent indications and further reading on wikipedia that the equation is the cartesian equation of an ellipse and that the exercise is a way of viewing the ellipse as an irregularly dilated or stretched circle. However, I don't see any rigorous, algebraic way of coming to that equation from the circle equation using the $F_{a,b}$ mapping, nor the other way around (e.g. using an inverse mapping $F^{-1}_{a,b}$). Applying the mapping to $u$ and $v$ on the left-hand side doesn't even seem to make sense. I might be missing something obvious...

Thanks in advance for your help.

PS: there isn't any development on ellipses in Lang's book (at least not in the part where the exercise comes from or before) so a solution can't rely on anything else that what was stated in the problem, except basic notions of coordinates, distance between points and Pythagora's theorem.

 

[Buffu]

Use parametrisation.

Let u = a sin t and v = b cos t.

 

[qspeechc]

To each point $(x, y)$ of the plane, associate the point $(ax, by)$.

Equation of a circle of radius 1 centered at the origin: $$x^2 + y^2 = 1$$

These are the relevant bits.

The first bit means, whenever you see $x$, write $ax$, whenever you see $y$ write $by$ (you can write it $x \to ax$ and $y \to by$ or something like that).

Now apply that to the second bit quoted above and take it from there.