$A(0,a)$ and $B(0,b),a,b 0$ are two vertices of a triangle $ABC$ where the vertex $C(x,0)$ is v

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$A(0,a)$ and $B(0,b),a,b > 0$ are two vertices of a triangle $ABC$ where the vertex $C(x,0)$ is variable. The value of $x$ when $\angle ACB$ is maximum is:

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[Key Concept: $\angle\ ACB$ is greatest when the circle through $A,B$ touches the $x$-axis at $C$; then ${OC}^{2} = OA \cdot \ OB$.]For $\angle ACB$ to be maximum, circle passing through $A,B$ will touch $x$-axis at $C$.$$\begin{matrix} OC^{2} & \ = (OA)(OB) \ x^{2} & \ = ab \ x & \ = \sqrt{ab} \end{matrix}$$

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