I believe the method of taking the averages between the two graphs can still applied using parametric equations, instead taking the averages of x and y individually. I have attached my solution using only parametric equations below. The equation for the y parameter on curve C may appear to be quite ambiguous as acos2(theta) + a and -acos2(theta) + a both give the same output. Please let me know if you have any further questions.
Where you say "straight line", I found it slightly confusing. I think here you mean that the (x,y) coordinates of the point R are given as (x,y)=(2a*cot(theta),2a) because the equation of line L is y=x*tan(theta) and it intersects y=2a when 2a=x*tan(theta) so x=2a*cot(theta).
Otherwise, I agree with curve C; I got the same and the parameterisation corresponds to point P hence justifying us taking the mipoint of PR for Q. I think the way you phrased it as "taking average between C and y=2a" threw me off a bit, but very good attempt nevertheless; many thanks!
Hi. Sorry for the late reply. I have attached my solution. Please do ask if anything seems confusing.
Hi. Sorry, that you haven't gotten a response yet. We will get to work on this right away!