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Circle, dilation and ellipse

  1. Oct 22, 2017 #1
    1. The problem statement, all variables and given/known data

    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}##."

    2. Relevant 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: $$x^2 + y^2 = 1$$

    3. The attempt at a solution

    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.
     
  2. jcsd
  3. Oct 22, 2017 #2
    Use parametrisation.

    Let u = a sin t and v = b cos t.
     
  4. Oct 24, 2017 #3
    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.
     
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