In a triangle the sum of two sides is x and the product of the same two sides is y. If x 2 - c 2 = y , (where c is the third side of the triangle) then the ratio of the inradius to the circumradius of the triangle is

Step-by-step solution

Let x = a + b y = a b x 2 - c 2 = y ⇒ a 2 + b 2 - c 2 2 a b = - 1 2 = cos ⁡ 120 o ⇒ ∠ C = 2 π 3 ⇒ R = a b c 4 ∆ , r = ∆ s ⇒ r R = 4 ∆ 2 s a b c = 4 1 2 a b sin ⁡ 2 π 3 2 x + c 2 y . c r R = 3 y 2 c x + c