71The expression readily simplifies to the factor if two expressions as :
(x+y-2006)(x+y+1) = 0
find the number of points (x,y) with positive integral coordinates satisfying the
x^2 + y^2 + 2xy - 2005x - 2005y - 2006 = 0.
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3 Answers
71
1\begin{align*} &\textup{we can factorise the given equation as,}\\ &(x+y)^2 -2005(x+y)-2006=0\\ &\textup{let x+y=t}\\ &\implies t^2-2005t-2006 =0 \\ &(t+1)(t-2006)=0 \\ & \implies \boxed{x+y = -1 / 2006} \\ & \textup{but x+y=-1 is ruled out as (x,y) are positive}\\ &= x+y=2006\\ & \textup{and since we are looking for positive integral solutions } \\ & \textup{therefore total \boxed{2005 solutions}} \end{align*}
