# Thread: Help with proving (ab'-a'b)^2+4(ah'-a'h)(bh'-b'h) is square of a rational number

1. ## Help with proving (ab'-a'b)^2+4(ah'-a'h)(bh'-b'h) is square of a rational number

Given:
a(x^2)+2hxy+b(y^2)=1
a'(x^2)+2h'xy+b'(y^2)=1

for x,y in R
where a,b,h,a',b',h' are rational.

Prove that:
i)(h-h')^2-(a-a')(b-b')
ii)(ab'-a'b)^2+4(ah'-a'h)(bh'-b'h)

are both squares of rational numbers.

I proved i) easily by simply subtracting the two equations and getting a quadratic in (x/y). I have been trying to prove ii) but don't know where to begin. If some one could give me a nudge in the right direction it will be very helpful.

Thanks

2. ## Re: Help with proving (ab'-a'b)^2+4(ah'-a'h)(bh'-b'h) is square of a rational number

Hi, please don't leave out conditions that is obvious to you, but not obvious to others.

Eg, I guess the given x,y equations are valid for any x,y in R, right? what is h? A constant in R?

3. ## Re: Help with proving (ab'-a'b)^2+4(ah'-a'h)(bh'-b'h) is square of a rational number

hi. sorry. Edited the post to answer your questions.