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 \(Q\).

1. Find the equation of \(L\).

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

3. Determine \(\big |QP\big |^2\) as a function of \(a\), where \(\big |QP\big |\) denotes the distance from \(P\) to \(Q\).

4. Find a point \(R\), in the \(xy\)-plane but not on \(C\), such that \(\big |RQ\big |\) is smallest for a unique value of \(a\). Briefly justify your answer.