Question

Two sides of a triangle are the roots of the equation $x^2-5x+3=0$. If the angle between the sides is $\frac{\pi }{3}$, then $r.R=$

• A. $\frac{4}{3}$
• B. $\frac{2}{3}$
• C. $\frac{2\sqrt{2}}{3}$
• D. $\frac{4\sqrt{2}}{3}$

Answer 1

Answer: (B)

$\frac{2}{3}$

If $a,b$ be the sides then $a+b=5,ab=3$
Also $\cos C=\frac{a^2+b^2-c^2}{2ab}$
or $\frac{1}{2}=\frac{{(a+b)}^2-2ab-c^2}{2ab}$
$\therefore 3=25-6-c^{2}or{c}^2=16\therefore c=4$
$\therefore r.R=\frac{S}{s}.\frac{abc}{4S}=\frac{abc}{2(a+b+c)}=\frac{3.4}{2(5+4)}=\frac{2}{3}$