Question:
In a $\Delta A B C$, if $\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}=k\left(a^{2}+b^{2}+c^{2}\right)$, then $k=$
Solution:
Using cosine formula
$\cos A=\frac{b^{2}+c^{2}-a^{2}}{2 b c}, \cos B=\frac{a^{2}+c^{2}-b^{2}}{2 a c}$ and $\cos C=\frac{a^{2}+b^{2}-c^{2}}{2 a b}$
L. H. S.
$=\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}$
$=\frac{b^{2}+c^{2}-a^{2}}{2 b c} \times \frac{1}{a}+\frac{a^{2}+c^{2}-b^{2}}{2 a c} \times \frac{1}{b}+\frac{a^{2}+b^{2}-c^{2}}{2 a b} \times \frac{1}{c}$
$=\frac{b^{2}+c^{2}-a^{2}+a^{2}+c^{2}-b^{2}+a^{2}+b^{2}-c^{2}}{2 a b c}$
$=\frac{a^{2}+b^{2}+c^{2}}{2 a b c}=k\left(a^{2}+b^{2}+c^{2}\right)$
$=$ R. H.S (given)
$\Rightarrow k=\frac{1}{2 a b c}$