JEE Advanced2016MathematicsProperties of TrianglesActual
In a ∆ X Y Z let x , y , z , be the lengths of sides opposite to the angles, X , Y , Z , respectively and 2 s = x + y + z . If s - x 4 = s - y 3 = s - z 2 and area of incircle of the triangle XYZ is 8 π 3 , then
Options
- AArea of the triangle XYZ is 6 6
- BThe radius of circumcircle of the triangle XYZ is 35 6 6
- Csin X 2 sin Y 2 sin Z 2 = 4 35
- Dsin 2 X + Y 2 = 3 5
Correct answer
A. Area of the triangle XYZ is 6 6
Step-by-step solution
Let s - x 4 = s - y 3 = s - z 2 = k ( K ≠ 0 ) s - x = 4 k s - y = 3 k s - z = 2 k Adding this, we get 3 s - x + y + z = 9 k ⟹ s = 9 k ⇒ x = 5 k , y = 6 k , z = 7 k ⇒ π r 2 = 8 π 3 π Δ s 2 = 8 π 3 ∆ = 8 3 . s s s - x s - y s - z = 8 3 . s 9 k . 4 k . 3 k . 2 k = 8 3 9 k 24.9 k 2 = 8 3 . 9 k ⇒ 3 × 2 6 K = 2 2 3 9 ⇒ K 2 2 . 9 6 6 3 = 3 2 3 2 = 1 k = 1 ⇒ x = 5 , y = 6 , z = 7 Δ = 8 3 . 9 k = 8 3 . 9 = 6 6 R = circumradius = x y z 4 Δ = 5 . 6 . 7 4 . . 6 6 = 35 4 6 Using formula sin X 2 sin Y 2 sin Z 2 = r 4 R = Δ