Question

In a triangle ABC, in usual notation, (a + b + c)(b + c – a) = λbc will be true if ______.

Options

• λ < 0

• λ > 0

• 0 < λ < 4

• λ > 4

Answer 1

In a triangle ABC, in usual notation, (a + b + c)(b + c – a) = λbc will be true if 0 < λ < 4.

Explanation:

Given (b + c + a)(b + c – a) = λbc

⇒ (b + c)² – a² = λbc

⇒ b² + c² – a² + 2bc = λbc

⇒ 2bc cos A + 2bc = λbc  (from cosine rule)

⇒ 2(cos A + 1) = λ

⇒ cos A = `λ/2 - 1`

But –1 < cos A < 1

⇒ `-1 < λ/2 - 1 < 1`

⇒ `0 < λ/2 < 2`

⇒ 0 < λ < 4