Question

If (a + b + c) (a $-$ b + c) = a² + b² + c². then show that a, b, c are in continued proportion.

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}: \\ \left(a+b+c\right)\left(a-b+c\right)=a^2+b^2+c^2 \\ \Rightarrow a^2-ab+ac+ba-b^2+bc+ca-cb+c^2=a^2+b^2+c^2 \\ =ac-b^2+ca=b^2 \\ =2ac=2b^2 \\ \therefore \frac{a}{b}=\frac{b}{c} \\ \mathrm{Hence},a,b\mathrm{and}c\mathrm{are}\mathrm{in}\mathrm{continued}\mathrm{proportion}. \end{gathered}$$