Question 604369
O(0,0) is the center,
(-4,0), and of course, symmetrically (4,0) are the co-vertices,
 so 4 is the semi-minor axis, b.
(0,3), and of course, symmetrically (0,-3) are the foci,
 so 3 is the focal distance, c.
The vertices are at (0,a) and (0,-a), with a being the semi-major axis and
{{{a^2=b^2+c^2=4^2+3^2=16+9=25=5^2}}}, so {{{highlight(a=5)}}}
and {{{x^2/b^2+y^2/a^2=1}}} so {{{x^2/3^2+y^2/5^2=1}}} or {{{highlight(x^2/9+y^2/25=1)}}}
 
HOW I KNOW THAT {{{a^2=b^2+c^2}}} :
In the diagram below, O is the center of the ellipse, with axes XA and BY.
(I did not draw the whole ellipse, just the important points).
{{{drawing(200,300,-4,4,-6,6,
line(0,-5,0,5),line(-3,0,3,0),
line(0,-4,3,0), line(3,0,0,4),
rectangle(0,0,0.4,0.4),
locate(3.1,0.3,B), locate(-3.3,0.3,Y),
locate(-0.2,5.6,A),locate(-0.2,-5.1,X),
locate(-0.4,4.3,C),locate(-0.4,-3.7,Z),
locate(0.1,0,O),locate(1.4,0,b),
arrow(-1,1.7,-1.0),arrow(-1,2.3,-1,4),
locate(-1.1,2.4,c),locate(-2.1,2.8,a),
arrow(-2,2.2,-2,0),arrow(-2,2.8,-2,5),
locate(1.8,2.2,a),locate(0.1,2,c),
locate(-3.5,-2,a^2=b^2+c^2)
)}}} A and X are vertices; B and Y covertices; C and Z foci.
The important segment lengths are:
OA=OX=a (the semi-major axis)
OB+OY=b (the semi-minor axis)
OC=OZ=c (the focal distance
Covertex B is one of the points of the ellipse.
The distances from B to focus C and to focus Z are the same BZ=BC.
Their sum BZ+BC=2BC is the same as the sum of distances to the foci for all points on the ellipse.
Vertex A is one of the points on the ellipse.
The sum of its distances to the foci is AC+AZ=AC+(AO+OZ)=(a-c)+(a+c)=2a
2BC=2a --> BC=a
In the right triangle OBC, Pythagoras theorem says that {{{a^2=b^2+c^2}}}