Question

Conic sections: Ellipse

Show that the distance between the directrix and center is D1C = a/e.

F1V1 + F1V2 = eD1V1 + eD1V2
2a = e(D1V1 + D1V2)
= e(D1V1 + [D1V1 + V1V2])
= e(D1V1 + V1V2 + V2D2)
= e(D1V1 + V1C + + CV2 + V2D2)
= e([D1V1 + V1C] + [CV2 + V2D2])
= e([D1C] + [CD2])
2a = 2eD1C
a/e = D1C

Prove that the distance from the center to the focus of an ellipse is given by F1C = ae.
(Hint: subtract instead of adding.)

N.B. All the '1's are subscript.