Question 1185877
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(Edited response.... I am not accustomed to the use of the term vertex when talking about ellipses, so I assumed the problem was supposed to have the center of the ellipse at the origin.  Doing that also results in a simpler equation for the ellipse, so possibly it was the intent of the problem that the origin is the center of the ellipse.  If that is not the case, then see the solution by tutor @ikleyn.)<br>
CENTER at the origin -- not the "vertex"<br>
General equations:<br>
(1) major axis horizontal: {{{(x-h)^2/a^2+(y-k)^2/b^2=1}}}<br>
(2) major axis vertical: {{{(x-h)^2/b^2+(y-k)^2/a^2=1}}}<br>
(h,k) is the center
a is length of semi-major axis
b is length of semi-minor axis<br>
In this example....
center is at the origin, so h=k=0
major axis is 10 so a=5
foci are on the x-axis, so major axis is horizontal<br>
That gives us, for the equation so far....<br>
{{{x^2/25+y^2/b^2=1}}}<br>
Complete the equation by finding the value of b^2, using the given point on the ellipse, (x,y)=(sqrt(5),2).<br>
{{{(sqrt(5))^2/25+(2^2)/b^2=1}}}
{{{5/25+4/b^2=1}}}
{{{4/b^2=1-1/5=4/5}}}
{{{b^2=5}}}<br>
ANSWER: {{{x^2/25+y^2/5=1}}}<br>