Question 1082287
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1.  The center of the ellipse is at (0,0).

    The major axis        is y-axis x= 0;
    Hence, the minor axis is x-axis y= 0.


2.  The canonical equation of this ellipse is 

    {{{x^2/b^2 + y^2/a^2}}} = 1    (it is written in the form to fit the fact that the major axis is y-axis: a > b > 0)

    Since the point (3,2) is on the ellipse, it implies 

    {{{3^2/b^2 + 2^2/a^2}}} = 1,   or

    {{{9a^2 + 4b^2}}} = {{{a^2*b^2}}}.    (1)


3.  The focal distance is 2c = 2 - (-2) = 4.
    Hence, the linear eccentricity c = {{{4/2}}} = 2. It means that

    {{{a^2 - b^2}}} = 4.          (2)


4.  Thus you have two equations (1) and (2) to determine "a" and "b".

    You can simplify writing and solving by introducing new variables x = {{{a^2}}} and y = {{{b^2}}}:

    9x + 4y = xy         (3)     instead of (1), and
    x - y = 4.           (4)     instead of (2)

The setup is done.
Now it is simple arithmetic to solve it and to get "a" and "b" at the end.


4.  From (4), x = 4 + y, Substitute it into (3). You will get

    9(4+y) + 4y = (4+y)*y,

    36 + 9y + 4y = 4y + y^2  --->  y^2 -9y - 36 = 0  ---->  {{{y[1,2]}}} = {{{(9 +- sqrt(81 + 4*36))/2}}} = {{{(9 +- 15)/2}}}.

    Only positive root works: y = 12.  So, {{{b^2}}} = 12 and b = {{{sqrt(12)}}}.

    Then a^2 = b^2 + 4 = 12 + 4 = 16 and  a = {{{sqrt(16)}}} = 4.

    Thus semi-axes are  4 (vertical) and {{{sqrt(12)}}} (horizontal).


    The equation for the ellipse is

    {{{x^2/16}}} + {{{y^2/12}}} = 1.
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As a prerequisite, see the lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Ellipse-definition--canonical-equation--characteristic-points-and-elements.lesson>Ellipse definition, canonical equation, characteristic points and elements</A> 

in this site.



Also, &nbsp;you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


The referred lesson is the part of this online textbook under the topic 
"<U>Conic sections: Ellipses. Definition, major elements and properties. Solved problems</U>".