Let a, b, c be the lengths of three sides of a triangle satistying the condition [(a^2+b^2…

Question

Let $a, b, c$ be the lengths of three sides of a triangle satistying the condition $$\left(a^2+b^2\right) x^2-2 b(a+c) x+\left(b^2+c^2\right)=0$$. If the set of all possible values of $x$ is the interval $$(\alpha, \beta)$$

Accepted Answer

$({a}^{2}+{b}^{2}){x}^{2}-2b(a+c)x+{b}^{2}+{c}^{2}=0$ $\begin{gathered} \Rightarrow {a}^{2}{x}^{2}-2abx+{b}^{2}+{b}^{2}{x}^{2}-2bcx+{c}^{2}=0 \ \Rightarrow (ax-b{)}^{2}+(bx-c{)}^{2}=0 \ \Rightarrow ax-b=0,\, bx-c=0 \ \Rightarrow a+b>c\, b+c>a\, c+a>b \end{gathered}$ $\begin{gathered} a+ax>bx \quad ax+bx>a \quad a{x}^{2}+a>ax \ a+ax>a{x}^{2} \quad ax+a{x}^{2}>a \quad {x}^{2}-x+1>0 \ {x}^{2}-x-10 \quad \end{gathered}$ $\begin{gathered} \frac{1-\sqrt{5}}{2}\frac{-1+\sqrt{5}}{2} \end{gathered}$ $\begin{gathered} \Rightarrow \frac{\sqrt{5}-1}{2}

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