If all solutions to a*x^2+2h*xy + b*y^2 = 1 and c*x^2+2g*xy + d*y^2 = 1 are rational, prove that...

Question

If all solutions to a*x^2+2h*xy + b*y^2 = 1 and c*x^2+2g*xy + d*y^2 = 1 are rational, prove that (ad-bc)^2+4(ag-ch)*(bg-dh) is the square of a rational

David · Accepted Answer

1. If all solutions to                    and                  are rational, prove 
that (     )   (     )(     ) is the square of a rational. 
Sol: 
Let(
 
 
)
 

 

 
   
The equations become,
We have three simultaneous equations, 
For a solution to exist for the first two equations,

If all solutions to ax^2+2hxy + by^2 = 1 and cx^2+2gxy + dy^2 = 1 are rational, prove that (ad-bc)^2+4(ag-ch)*(bg-dh) is the square of a rational

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