Parametric Form in Conics (1 Viewer)

[beanbean]

Hi there friends,
I've been doing this question:
P is a point on the ellipse with equation x^2/a^2 +y^2/b^2 = 1 with focus at S.
The normal at P intersects the X-axis at Q. Show that QS=ePS.

I let my co-ordinates by (x1, y1), found the point Q and distance of PS.
I then tried to prove QS = ePS using LHS & RHS.
LHS went fine, but RHS I could not figure out!!
Looking at the solutions our teacher did, he used (acos(theta), bsin(theta)) and he proved QS = ePS
Are we meant to always use parametric form for questions where no actual numerical values are provided??
Please help! Any help is appreciated tbh!!
Thanks for even looking

 

[braintic]

Hi there friends,
I've been doing this question:
P is a point on the ellipse with equation x^2/a^2 +y^2/b^2 = 1 with focus at S.
The normal at P intersects the X-axis at Q. Show that QS=ePS.

I let my co-ordinates by (x1, y1), found the point Q and distance of PS.
I then tried to prove QS = ePS using LHS & RHS.
LHS went fine, but RHS I could not figure out!!
Looking at the solutions our teacher did, he used (acos(theta), bsin(theta)) and he proved QS = ePS
Are we meant to always use parametric form for questions where no actual numerical values are provided??
Please help! Any help is appreciated tbh!!
Thanks for even looking

Either method is fine. See my solution for Cartesian coords:

View attachment Soln.pdf