JEE Main · 2019 · 11 Jan 2019 · Shift 1 · Properties of Triangle

In a triangle, the sum of lengths of two sides is

x

and the product of the lengths 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:

Answer: (B)

Solution

Let the two sides be

a

and

b

with

a + b = x

and

ab = y

. The condition

x^{2} - c^{2} = y

reads

(a + b)^{2} - c^{2} = ab

, so

a^{2} + b^{2} - c^{2} = - ab

. The cosine rule then gives

cos C = \frac{a ^{2} + b ^{2} - c ^{2}}{2 ab} = \frac{- ab}{2 ab} = - \frac{1}{2}

, so

C = 12 0^{\circ}

and

sin C = \frac{3}{2}

. By the sine rule the circumradius is

R = \frac{c}{2 sin C} = \frac{c}{3}

JEE Main · 2019 · 11 Jan 2019 · Shift 1 · Single correct option