Dancing On a Parabola

Calculus Level 2

Suppose that two people, A and B, walk along the parabola \(y=x^2 \) in such a way that the line segment \( L\) between them is always perpendicular to the line tangent to the parabola at A's position \((a,a^2)\) with \( a > 0 \). If B's position is \((b,b^2)\), what value of \(b\) minimizes \(L\)?

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