Geometric/algebraic proof of a quadratic

[raphael3d]

http://img717.imageshack.us/img717/4029/screenshot20110106at123.png

i don't know how to construct an algebraic proof from this or how to attempt it.
is there anything obvious which i missed?


thank you

 

[verty]

I think the question is badly put. By varying the value of c, one can ensure that P'Q' does not cut the circle. So there would be no M and N.

Perhaps they mean that if there is an M and N, then QX and QY will be the roots of the equation.

One way you could solve it is:
- find the coordinates of N and M, assuming they exist
... rest removed.

 

[raphael3d]

one can then also vary P' and Q' so they do cut the circle at points M and N, so the construction seems consistent.

its an example from "course in pure mathematics" from g.h.hardy

how should one find points like M and N, if only lengths are given and asked?
this is no function space, just geometry.

for instance: QN/2=NY/QY, but how could someone proceed then?

 

[verty]

If you know how to solve it with coordinate geometry, you can translate the proof to the form you require. For example, using the gradient of P'Q' is possible without coordinates.