Question

If α₁, α₂ are the roots of equation x² – px + 1 = 0 and β₁, β₂ be those of equation x² – qx + 1 = 0 and vector α₁$\widehat{i}$ + β₁$\widehat{j}$is parallel to α₂$\widehat{i}$+ β₂$\widehat{j}$ then –

• A. p = ± q
• B. p = ± 2q
• C. p = 2q
• D. None of these

Answer 1

Answer: (A)

p = ± q

Answer 2

α₁ + α₂ = p β₁ + β₂ = q

α₁α₂ = 1 β₁β₂ = 1

Let $\overset{\rightarrow}{a} = \alpha_{1}\widehat{i} + \beta_{1}\widehat{j}$ and $\overset{\rightarrow}{b} = \alpha_{2}\widehat{i} + \beta_{2}\widehat{j}$are parallel

⇒ $\frac{\alpha_{1}}{\alpha_{2}}$=$\frac{\beta_{1}}{\beta_{2}}$ ⇒ $\frac{\alpha_{1}\alpha_{2}}{(\alpha_{1} + \alpha_{2})^{2}}$ = $\frac{\beta_{1}\beta_{2}}{(\beta_{1} + \beta_{2})^{2}}$

⇒ $\frac{1}{p^{2}}$ = $\frac{1}{q^{2}}$ ⇒   p² = q² ⇒ p = ± q