Ellipses and eccentricity

Hi, can someone explain when I would need to use the property PS = ePM
It crops up in part c of the question but I didn’t know how to do it. e is root(3) /2
I took a general point P (x, y) and set the distance of it from the foci as 2

Naively youd use tthe equation (part a) and pythagoras/circle from part b) to solve a couple of quadratic equations?

Yes, that’s what I started to do

It would help to see what youve done / what youve had a problem with when you post the question, but if you started doing that, what was the problem, why do you think you should use PS = ePM ...?

I did it this way But the mark scheme uses PS = ePM so I was wondering why that works / when I need to use this / what cues are in the question that suggest I can use this

From looking at the mark scheme (it would have been helpful to post it), the alternative method is what you did. The main method uses the focus-directrix property so for any point on an ellipse, the ratio of its distance to a focus and its distance to its corresponding directrix is constant (equal to the eccentricity). I suppose as the question is based on specifying the ellipse in terms of the directrices, and as it avoids solving quadratic simultaneous equations, it is preferable. Similarly, 5 marks is a bit stingy to do simultaneous quadratic equations (sort of twice and evaluate y). One to remember if youre given an ellipse and directrices.

Thanks. I also did a similar thing for this Q but again they used PS = ePM
Why does it work here