The equation of the ellipse that has a center at (0,0) , a focus at (3 , 0) , and a vertex at (5 , 0 ) , is (x^2)/(A^2)+(y^2)/(B^2)=1
where A =____
B =_______
The equation is given by :
x ^2 / 25 + y ^2 / 16
So ... A = 5 and B = 4
Here's the graph.......https://www.desmos.com/calculator/pfvhokujae
http://www.mathopenref.com/coordgeneralellipse.html
If the centre is (0,0)
Let the end of the horizontal axis is A(5,0)
The foci is $$S_1(3,0)$$
If you sketch this it is easy to see that the other foci is $$S_2(-3,0)$$
Let the point B(0,b) be the end of the vertical axis.
NOW
$$\\S_1B=BS_2
S_1A+AS_2\;\; must\; \;equal \;\; S_1B+BS_2\\\\
2+8=S_1B+BS_2\\\\
10=5+5\\\\
so\\\\
B(0,4)\\\\
$equation of the ellipse$\\\\
\frac{x^2}{5^2}+\frac{y^2}{4^2}=1\\\\
\frac{x^2}{25}+\frac{y^2}{16}=1$$