Question 1130622
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An ellipse has two axes; and there are two coordinate axes.  So when the statement of the problem sloppily says "the" axis of the ellipse coincides with "the" coordinate axis, I will assume that the intended meaning is that both axes of the ellipse coincide with both coordinate axes -- i.e., the center of the ellipse is at the origin.<br>
Then the equation of the ellipse is <br>
{{{x^2/a^2+y^2/b^2 = 1}}}<br>
Form two equations in a^2 and b^2 using the two known points on the ellipse and solve the pair of equations to find the equation of the ellipse.<br>
{{{9/a^2+25/b^2 = 1}}}
{{{49/a^2+(25/9)/b^2 = 1}}}<br>
Multiply the second equation by 9 and subtract one equation from the other to eliminate b:<br>
{{{441/a^2+25/b^2 = 9}}}
{{{432/a^2 = 8}}}
{{{8a^2 = 432}}}
{{{a^2 = 54}}}<br>
Substitute in either original equation to find b^2:<br>
{{{9/54+25/b^2 = 1}}}
{{{25/b^2 = 1-9/54 = 1-1/6 = 5/6}}}
{{{5b^2 = 150}}}
{{{b^2 = 30}}}<br>
We nowknow a^2 and b^2 for the ellipse, so we can write its equation:<br>
ANSWER: {{{x^2/54+y^2/30 = 1}}}