A Vector Combination on a Circle

★ 8/10Extrema Circles Estimation inequality vectors

Problem

Points $T, M, N$ are three distinct points on the circle $C: (x-1)^2 + y^2 = 4$ , and

\overrightarrow{CT} = a\,\overrightarrow{CM} + b\,\overrightarrow{CN}

for positive reals $a, b$ . Find the range of

\frac{a^3 + ab^2 + 2ab + b + 1}{a}.

Answer

Solution

Difficulty	8/10
Topics	Extrema, Circles, Estimation, inequality, vectors

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