Question

$\frac{\sqrt{a} + \sqrt{x - b}}{\sqrt{b} + \sqrt{x - a}} = \frac{\sqrt{a}}{\sqrt{b}} = \frac{\sqrt{x - b}}{\sqrt{x - a}}$

$$\begin{array}{r} \begin{matrix} \Rightarrow & ax - a^{2} = bx - b^{2} & \\ \Rightarrow & (a - b)(x - (a + b)) = 0 & \\ \therefore & x = a + b & \text{~if~} \\ & x \in \lbrack a,\infty) & \text{~if~} \\ a > 0,b^{2} \leq 4ac & & a = b \\ b > 0,c^{2} \leq 4ab & & \\ c > 0,a^{2} \leq 4bc & & \\ \Rightarrow & a^{2} + b^{2} + c^{2} < 4(ab + bc + ca) & \lbrack\because \\ \text{~Also~} & a^{2} + b^{2} + c^{2} > ab + bc + ca & \\ \Rightarrow & \frac{a^{2} + b^{2} + c^{2}}{ab + bc + ca} \in (1,4) & (\because a,b,c\text{~are distinct~}) \end{matrix}\#(12.) \end{array}$$