There exist nonzero integers $a$ and $b$ such that the quadratic
\[(ax - b)^2 + (bx - a)^2 = x\]has one integer root and one non-integer root.  Find the non-integer root.
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<|im_start|>user
There exist nonzero integers $a$ and $b$ such that the quadratic
\[(ax - b)^2 + (bx - a)^2 = x\]has one integer root and one non-integer root.  Find the non-integer root.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
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user
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There exist nonzero integers $a$ and $b$ such that the quadratic
\[(ax - b)^2 + (bx - a)^2 = x\]has one integer root and one non-integer root.  Find the non-integer root.
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