When does the normal to a parabola pass through this point?

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As shown in the diagram below: \(C\) is the parabola with equation \(y=x^2\); \(P\) is the point \((0,1)\); \(Q\) is the point \((a,a^2)\) on \(C\); \(L\) is the normal to \(C\) which passes through \(Q\).

2. For what values of \(a\) does \(L\) pass through \(P\)?

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