blog.bernatchez.net

The following figure shows part of the graph of \(f(x) = 2x\sqrt[2]{a^2 - x^2}\), for \(-1 \leq x \leq a\), where \(a > 1\).

The figure is not to scale.

The line \(L\) is the tangent to the graph of \(f\) at the origin, \(O\). The point \(P(a; b)\) is on \(L\).

1. Given that \(f^\prime(x) =\frac{2a^2 - 4x^2}{\sqrt{a^2-x^2}}\), for \(-1 \leq x \leq a\), find the equation of \(L\).
2. Hence or otherwise, find an expression for \(b\) in terms of \(a\).