Question 1101761
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The coordinates of the two foci are (1,-7) and (1,-1).  That tells us...
(1) The center of the ellipse is (1,-4);
(2) the major axis is vertical; and
(3) the distance from the center to each focus is 3.<br>
The standard form for the equation of an ellipse with a vertical major axis is<br>
{{{((x-h)^2/b^2) + ((y-k)^2/a^2) = 1}}}<br>
(h,k) is the center of the ellipse ((1,-4) in this example);
a is the semi-major axis (5 in this example);
b is the semi-minor axis;
and the values of a and b are related by the equation {{{a^2-b^2 = c^2}}}, where c is the distance from the center of the ellipse to each focus (3 in this example).<br>
In this example, since a is 5 and c is 3, b is 4.<br>
And now we have all the specific values needed to write the equation of this ellipse:<br>
{{{((x-1)^2/4^2) + ((y+4)^2/5^2) = 1}}}<br>
or<br>
{{{((x-1)^2/16) + ((y+4)^2/25) = 1}}}<br>