Question

If P is a point on an ellipse with a focus F, then PF is always greater than PM,
where M is the foot of the perpendicular drawn from P to a directrix of the ellipse.

Answer 1

The statement is false.

If F is a focus of the ellipse, P is a point on the ellipse and M is foot of perpendicular from P on the directrix of the ellipse.

Then the distances PM and PF form a known ratio, which is $PF/PM=e$ ;

where e is eccentricity of the ellipse $\implies 0<e<1$ .

Thus, $0<PF/PM<1 \implies PF<PM$ .

Hence, PF is always less than PM.