Solve the system of equations \begin{align}x + y + z &= a\\ x^2 + y^2 + z^2 &= b^2\\ xy &= z^2,\end{align} where $a$ and $b$ are constants. Give the condition on $a$ and $b$ so that $x,y$ and $z$ are distinct. I solved for $x,y,z$. But I couldn't understand how to impose conditions on them to make them unique.
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1What is your solution for $x,y,z$ ? – Emilio Novati Jul 11 '17 at 07:38
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I solved it and got z = (a^2 - b^2) / 2a and x and y I got really very unusual answer in terms of a and b – saisanjeev Jul 11 '17 at 07:45
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$x,y,z$ distinct means they are different $x\ne y\land x\ne z\land y\ne z$? – Raffaele Jul 11 '17 at 11:23
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HINT: plugging $$z=a-x-y$$ in the second and third equation we obtain: $$a^2-2ax-2ay-b^2+2x^2+2xy+2y^2=0$$ (I) $$-a^2+2ax+2ay-x^2-xy-y^2=0$$ (II) multiplying (II) by $2$ and adding to (I) we get $$-a^2+2ax+2ay-b^2=0$$(III) this equation is linear in $x,y$ and you can use this to compute the other variables.
Dr. Sonnhard Graubner
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This would help me to get the values of x,y, and z , but how do i prove the second part of the question – saisanjeev Jul 11 '17 at 09:16
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Yes they are quadratic , but I do get some values(they are very long expressions,though), but how do I impose the condition for a,b to make them unique? – saisanjeev Jul 11 '17 at 09:25
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