Question

In a ∆ABC, if $\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}=k\left(a^2+b^2+c^2\right),$ then k = ___________________.

Answer 1

Using cosine formula

$$\begin{gathered} \cos A=\frac{b^2+c^2-a^2}{2bc},\cos B=\frac{a^2+c^2-b^2}{2ac}\mathrm{and}\cos C=\frac{a^2+b^2-c^2}{2ab} \\ \mathrm{L}.\mathrm{H}.\mathrm{S} \\ =\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c} \\ =\frac{b^2+c^2-a^2}{2bc}\times \frac{1}{a}+\frac{a^2+c^2-b^2}{2ac}\times \frac{1}{b}+\frac{a^2+b^2-c^2}{2ab}\times \frac{1}{c} \\ =\frac{b^2+c^2-a^2+a^2+c^2-b^2+a^2+b^2-c^2}{2abc} \\ =\frac{a^2+b^2+c^2}{2abc}=k\left(a^2+b^2+c^2\right) \\ =\mathrm{R}.\mathrm{H}.\mathrm{S}\left(\mathrm{given}\right) \\ \Rightarrow k=\frac{1}{2abc} \end{gathered}$$