A Distance in Disguise

★ 7/10trigonometry Cauchy-Schwarz analytic geometry

Problem

Find the maximum value of

f(t, \alpha) = \frac{\left|\left(\cos\alpha + \sqrt{2}\sin\alpha\right)t - \sqrt{2}\right|}{\sqrt{t^2 - 2\sqrt{2}\,t\cos\alpha + 2}},

where $t \in \mathbb{R}$ and $\alpha \in \left(0, \dfrac{\pi}{2}\right)$ .

Answer

Solution

Difficulty	7/10
Topics	trigonometry, Cauchy-Schwarz, analytic geometry

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