Question

Let $a$, $b$ and $c$ be the sides of a $\Delta ABC$. If $a^2,b^{2}and{c}^2$ are the roots of the equation $(x^3-P{x}^2+Qx-R=0)$, where P, Q and R are constants then the value of

$(\frac{cosA}{a}+\frac{cosB}{b}+\frac{cosC}{c})$ is equal to

• A. $\frac{P}{2\sqrt{R}}$
• B. $\frac{P}{\sqrt{2R}}$
• C. $\frac{P\sqrt{2}}{R}$
• D. none of these

Answer 1

Answer: (A)

$\frac{P}{2\sqrt{R}}$

Given

$a^2,b^2,c^2$ are roots of $x^3-P{x}^2+Qx-R=0$

$⟹a^2+b^2+c^2=P$

$⟹a^2{b}^2{c}^2=R$

$⟹abc=\sqrt{R}$

Consider $\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}$

$⟹\frac{b^2+c^2-a^2}{2abc}+\frac{a^2+c^2-b^2}{2abc}+\frac{b^2+a^2-c^2}{2abc}$

$(\cos A=\frac{b^2+c^2-a^2}{2bc})$

$⟹\frac{a^2+b^2+c^2}{2abc}$

$⟹\frac{P}{2\sqrt{R}}$