In a triangle, the sum of lengths of two sides is x and the product of length of the same two sides is y. if $x^{2} - c^{2} = y$ , where c is the length of the third side of the triangle, then the circumradius of the triangle is _______

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$\frac{C}{\sqrt{3}}$

$\frac{y}{\sqrt{3}}$

$\frac{3 y}{2}$

$\frac{C}{3}$

answer is B.

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Detailed Solution

$\Rightarrow (a + b)^{2} - c^{2} = a b a^{2} + b^{2} - c^{2} + 2 a b = a b a^{2} + b^{2} - c^{2} = - a b \frac{a^{2} + b^{2} - c^{2}}{2 a b} = \frac{- 1}{2} \cos C = \cos \frac{2 π}{3} C = \frac{2 π}{3} R = \frac{c}{2 \sin c} = \frac{c}{2 \times \sin \frac{π}{3}} = \frac{c}{2 \times \frac{\sqrt{3}}{2}} = \frac{C}{\sqrt{3}}$