Construction of an ellipse Is it possible to construct an ellipse with a line, compasses and a pencil? If yes, how and why is the construction correct?
 A: As said in the comments, this is not possible. The ellipse consists of a line which is neither a straight line (ruler) nor some arc of a circle (compass). It is however possible to draw an ellipse using two pins, a piece of rope and a pencil.
Pin one end of the rope to the paper and pin the other end on the paper as well, in such a way that the rope is not taut. Then pull the rope taut using the pencil and move the pencil around (while keeping the rope taut) to draw an ellipse.

$F_1, F_2$ are the foci of the ellipse and the length of the string ($a+b$) is of course constant and equal to the length $PQ$ (the long axis).
A: You can construct an arbitrary number of points on an ellipse using e.g. Pascal's theorem (which doesn't even require compasses). If you want the ellipse as a curve, though, not as a set of points, then this is strictly speaking impossible since both ruler and compasses can only draw curves of constant curvature, which the ellipse is not. You can, of course, come arbitrary close.
A: here is a compass and rule construction of of an ellipse with major axis $a$ and foci at $F_1 = (-ae, 0)$ and $F_2 = (ae, 0)$
here are the construction steps:
(a) pick two points $F_1, F_2.$ draw the line $F_1F_2$
(b) draw a circle with center $F_1$ and radius big enough to contain $F_2$ within it. let the circle cut $F_1F_2$ extended at $A$ so that $F_2$ is between $F_1$ and $A.$
(c) pick any point $Q$ on the circle, draw the perpendicular bisector of $F_2Q$.
  let the bisector cut $F_1Q$ at $P.$
(d) $P$ is on required ellipse. now move the point $Q$ on the circle and get more of the points $P$ corresponding to $Q.$
