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Finding The Equation of an Ellipse

  1. Mar 8, 2012 #1
    1. The problem statement, all variables and given/known data

    The point P on a circle is transformed into the point P' on a ellipse.

    The point P is (6,8), and lies on a circle with the equation X^2 + Y^2 = 100.
    Point P' lies on the same graph after it has been transformed into a ellipse, with the co-ordinates (4,12). (No transitions, only a stretch factor change.) Both conics have an origin of 0,0. Find the equation for the new ellipse in the form, x^2 / a^2 + y^2 / b^2 = 1. Most importantly find the stretch factors A and B.



    2. Relevant equations

    standard circle = x^2 + y^2 = r^2
    standard ellipse = x^2 / a^2 + y^2 / b^2 = 1.


    3. The attempt at a solution

    Till this point, I have only done this type of question when the points being transformed lied on either the major or minor axis. I have attempted to substitute the point in which the ellipse passes through, but to no avail, as I cannot solve for both A and B at the same time as they're both unknown. I have thought about linear systems, but I only have one pair of co-ordinates, and I don't believe I can figure out the major/minor axis given my level of trigonometry (only works on circles). I am unable to identity the relationship between P and P' and its relevance to helping me find A and B.
     
  2. jcsd
  3. Mar 8, 2012 #2

    jambaugh

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    Science Advisor
    Gold Member

    Since both conics have center at (0,0), all points on the circle scale by the same x factor and by the same y-factor as they are transformed into the ellipse.

    What horizontal scaling factor will map the x coordinate of P to the x coordinate of P'?
    What vertical scaling factor will map the y coordinate of P to the y coordinate of P'?

    On a relevant but general note. If you have a relation expressed as an equation or inequality in x and y. Then a horizontal scaling by a factor of a and a vertical scaling by a factor of b is accomplished by the variable substitution x--> x/a, y-->y/b.

    Similarly a horizontal translation by a distance of h and vertical translation by a distance of k results from/is accomplished by the substitution x --> (x-h), y--> (y-k).

    With that in mind let me give one more hint. A circle is an ellipse and so can be written in the standard ellipse form: (divide through by r^2).
     
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