Question 1119425
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With foci at (-4,0) and (4,0), the center of the ellipse is at (0,0).  The equation is of the form<br>
{{{x^2/a^2+y^2/b^2 = 1}}}<br>
The distance from the center of the ellipse to each focus is c, where {{{c^2 = a^2-b^2}}}<br>
Since that distance is 4, we have<br>
{{{a^2-b^2 = c^2 = 16}}}
{{{a^2 = b^2+16}}}<br>
Then the equation of the ellipse can be written as<br>
{{{x^2/(b^2+16) + y^2/b^2 = 1}}}<br>
We can find the value of b^2 by plugging in the coordinates of the known point on the ellipse, (4,1).<br>
{{{16/(b^2+16)+1/b^2 = 1}}}
{{{16(b^2)+(b^2+16) = b^2(b^2+16)}}}
{{{17b^2+16 = b^4+16b^2}}}
{{{b^4-b^2-16 = 0}}}
{{{b^2 = (1+sqrt(65))/2}}}<br>
Then<br>
{{{a^2 = b^2+16 = (1+sqrt(65))/2+16 = (33+sqrt(65))/2}}}<br>
And finally the equation is<br>
{{{x^2/((33+sqrt(65))/2)+y^2/((1+sqrt(65))/2) = 1}}}