Question

In a∆ABC, a, b, A are given and c₁, c₂ are two values of the third side c. The sum of the areas of two triangles with sides a, b_, c₁ and a, b, c₂ is

• A. $\frac { 1 } { 2 } b ^ { 2 } \sin 2 A$
• B. $\frac { 1 } { 2 } a ^ { 2 } \sin 2 A$
• C. b² sin2A
• D. None

Answer 1

Answer: (A)

$\frac { 1 } { 2 } b ^ { 2 } \sin 2 A$

Answer 2

We have cos A =

⇒c² – 2bc cos A + b² – a² = 0.

It is given that c₁ and c₂ are the roots of this equation.

Therefore c₁ + c₂ = 2b cos A and c₁c₂ = b² – a². Now sum of the areas of the two triangles = ½ c₁b sin A+ ½ c₂b sin A

= ½ (c₁ + c₂)b sin A = b. b sin A cos A = ½ b² sin2A