$A, B$ are fixed points with coordinates $(0, a)$ and $(0, b)(a 0, b 0)$. $P$ is variable point $(x, 0)$ referred to rectangular axis. If the angle $\angle \mathrm{APB}$ is maximum, then

Question

$A, B$ are fixed points with coordinates $(0, a)$ and $(0, b)(a > 0, b > 0)$. $P$ is variable point $(x, 0)$ referred to rectangular axis. If the angle $\angle \mathrm{APB}$ is maximum, then, $x^2=a b$, $x^2=a+b$, $x=\frac{1}{a b}$, $x=\frac{a+b}{2}$

MARKS App · Accepted Answer

Hint : $\angle A P B=\theta=\cos ^1\left(\frac{x^2+a^2+x^2+b^2-(a-b)^2}{2 \sqrt{x^2+a^2} \sqrt{x^2+b^2}}\right)$ For Max; $\frac{\mathrm{d} \theta}{\mathrm{dx}}=0, \mathrm{x}^2=\mathrm{ab}$

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