The standard equation for an ellipse is:
x2/a2+y2/b2=1 where a and b are the semi-axes. In this case a2=1 and b2=4, making a=1 and b=2, so b=a and b is the semi-major axis (vertical) and a is the semi-minor axis (horizontal).
The ellipse has its centre at the origin (0,0) so the vertices are at (-a,0), (a,0), (0,-b), (0,b), that is, (-1,0), (1,0), (0,-2), (0,2).
Every point P on the ellipse is positioned so that PF1+PF2=constant, where F1 and F2 are the foci. In this case the foci are both on the y-axis and if c is the focal length then the foci are at (0,c) and (0,-c) because the ellipse is centrally located.
If we take a vertical vertex we can use geometry to calculate c. At P(0,2), PF1=2-c and PF2=2+c so:
PF1+PF2=2-c+2+c=4. 4 is 2b in this case.
Therefore, if we take the horizontal vertices, P(1,0) we get an isosceles triangle (in this case, the isosceles triangle is actually an equilateral triangle), with a focus at the apex (0,c) and the other two vertices at (-1,0) and (1,0). The length of equal sides of the triangle can be found using Pythagoras:
√(a2+c2)=√(1+c2). So the sum of these equal sides is 2√(1+c2)=4, √(1+c2)=2, 1+c2=4, c2=3, so focal length c=√3, and the foci are at (0,√3) and (0,-√3).