Question

In a DABC, if a, b, c and A are given, then there are two D’s with third sides c₁ and c₂, then (c₁ – c₂) is –

• A. Ab sin A
• B.
• C. $2 \sqrt { a ^ { 2 } - b ^ { 2 } \sin ^ { 2 } A }$
• D. 2ab sin A

Answer 1

Answer: (C)

$2 \sqrt { a ^ { 2 } - b ^ { 2 } \sin ^ { 2 } A }$

Answer 2

a² = b² + c² – 2bc cos A

̃ c² – 2bc cos A + (b² – a²) = 0

Clearly c₁ and c₂ are roots of equation

c₁ + c₂ = 2b cos A, c₁c₂ = b² – a²

\ c₁ – c₂ =

= $\sqrt { 4 b ^ { 2 } \cos ^ { 2 } A - 4 \left( b ^ { 2 } - a ^ { 2 } \right) }$

=

= $2 \sqrt { a ^ { 2 } - b ^ { 2 } \sin ^ { 2 } A }$ .