Question

let a,b,c be three distinct real numbers such that each of the expression ax^​2+bx+c,bx²+cx+aand cx²+ax+b is positive for each x belongs to R and let alpha = (bc+ca+ab)/(a²+b²+c²);
(A) alpha <4 (B) alpha <1 (C) alpha >1/4 (D) alpha >1

Answer 1

$$\begin{gathered} {\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}>0\forall \mathrm{x}\in \mathbf{R} \\ \mathbf{\Rightarrow }\mathrm{a}>0\mathrm{and}{\mathrm{b}}^2-4\mathrm{ac}<0...\left(1\right) \\ {\mathrm{bx}}^2+\mathrm{cx}+\mathrm{a}>0\forall \mathrm{x}\in \mathbf{R} \\ \mathbf{\Rightarrow }\mathrm{b}>0\mathrm{and}{\mathrm{c}}^2-4\mathrm{ab}<0...\left(2\right) \\ {\mathrm{cx}}^2+\mathrm{ax}+\mathrm{b}>0\forall \mathrm{x}\in \mathbf{R} \\ \mathbf{\Rightarrow }\mathbf{c}>0\mathrm{and}{\mathrm{a}}^2-4\mathrm{bc}<0...\left(3\right) \\ \mathrm{Adding}\left(1\right),\left(2\right)\mathrm{and}\left(3\right),\mathrm{we}\mathrm{get} \\ \therefore {\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2-4\mathrm{ab}-4\mathrm{bc}-4\mathrm{ca}<0 \\ \Rightarrow {\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2<4\mathrm{ab}+4\mathrm{bc}+4\mathrm{ca} \\ \Rightarrow \frac{\mathrm{ab}+\mathrm{bc}+\mathrm{ca}}{{\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2}>\frac{1}{4} \\ \Rightarrow \mathrm{\alpha }>\frac{1}{4} \end{gathered}$$