Question

Each pair of equations from x² – b_rx + c_r = 0, r = 1, 2, 3 have a common root and the relation is given Σ$b_{1}^{2}$ + 4 Σc₁ = k Σb₁b₂ then value of k is –

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (B)

2

Answer 2

Equations x² – b₁ x + c₁ = 0 roots α,β

x² – b₂ x + c₂ = 0 roots α,γ

x² – b₃ x + c₃ = 0 roots β,γ

∴ α + β = b₁, αβ = c₁ 

α + γ = b₂, αγ = c₂

β + γ = b₃, βγ = c₃

L.H.S  = $b_{1}^{2}$+ $b_{2}^{2}$ + $b_{3}^{2}$+ 4(c₁ + c₂ + c₃)

= (α + β)² + (α + γ)² + (β + γ)² + 4 (αβ +βγ + αγ)

= 2[α² + β² + γ² + 3(αβ +βγ + αγ)]

R.H.S = k[b₁b₂ + b₂b₃ + b₃b₁]

= 2(α + β) (α + γ) + (α + γ) (β +γ) +(β + γ)(α + β)

= 2[α² + β² + γ² + 3(αβ +βγ + αγ)]

⇒ k = 2