Question

In $\Delta$ $ABC$ the sides opposite to angles $A,B,C$ are denoted by $a,b,c$ respectively.

Then the $\frac{a^2+b^2+c^2}{R^2}$ has the maximum value of?

• A. $3$
• B. $6$
• C. $9$
• D. none of these

Answer 1

Answer: (C)

$9$

$a=2RsinA$
$b=2RsinB$
$c=2RsinC$

Hence $\frac{a^2+b^2+c^2}{R^2}$

$=\frac{4R^2(si{n}^{2}A+si{n}^{2}B+si{n}^{2}C)}{R^2}$
$=4(si{n}^{2}A+si{n}^{2}B+si{n}^{2}C)$

Now it is maximum for $sinA=sinB=sinC$ ... (equilateral triangle).
$=4\left(3si{n}^2{60}^0\right)$
$=12\times \frac{3}{4}$
$=9$