Question

In a triangle ABC, the sides $b$ and $c$ are the roots of the equation $x^2-61x+820=0$ and $A=ta{n}^{-1}\left(4/3\right)$, then $a^2$ is equal to

Answer 1

Given sides b and c are roots of the equation
$x^2-61x+820=0$
$\Rightarrow (x-41)(x-20)=0$
$\Rightarrow x=41,20$
$\Rightarrow b=41,c=20$
Also given $\tan A=\frac{4}{3}$
$\Rightarrow \cos A=\frac{3}{5}$
Now, $a^2=b^2+c^2-2bc\cos A$
$=1681+400-2\times 41\times 20\times \left(\frac{3}{5}\right)=1097$