Question 1194678
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What is the equation that represents an ellipse with foci at (3,2) and (-9,2) 
that passes through the point (-3,10)? Show all steps and provide the answer 
(the ellipse's equation).
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<pre>
From the given data, the major axis is horizontal line y = 2.


The center of the ellipse is the point ({{{(3 + (-9))/2}}},{{{2}}}) = (-3,2).


The focal distance between the foci is 3+9 = 12 units;
the eccentricity "c" (half of the focal distance) is c = 12/2 = 6 units.


Since the ellipse passes through the point (-3,10) with the same x-coordinate 
as the center has, it implies that the minor semi-axis "b" of the ellipse 
has the length of  b = 10 - 2 = 8.


Hence, the major semi-axis "a" is  a = {{{sqrt(b^2 + c^2)}}} = {{{sqrt(8^2 + 6^2)}}} = {{{sqrt(100)}}} = 10.


Now the equation of the ellipse is

    {{{(x+3)^2/a^2}}} + {{{(y-2)^2/b^2}}} = 1,

or

    {{{(x+3)^2/10^2}}} + {{{(y-2)^2/8^2}}} = 1.    <U>ANSWER</U>
</pre>

Solved.