Question

If α, β are roots of x² – 3x + 1 = 0 then the equation whose roots are $\frac{1}{\alpha –2}$, $\frac{1}{\beta –2}$ is –

• A. x² + x – 1 = 0
• B. x² + x + 1 = 0
• C. x² – x – 1 = 0
• D. None of these

Answer 1

Answer: (C)

x² – x – 1 = 0

Answer 2

α, β → x² – 3x + 1 = 0 ; α/β = x

$\frac{1}{\alpha –2}$or $\frac{1}{\beta –2}$= x ⇒ (α – 2) or (β – 2) = $\frac{1}{x}$

x = α /β = $\frac{1}{x}$ + 2 ⇒  x →$\frac{1}{x}$ + 2

Required quadratic equation

= $\left( \frac{1}{x} + 2 \right)^{2}$–3$\left( \frac{1}{x} + 2 \right)$ +1 = 0

⇒ 4x² + 4x + 1 – 3x – 6x² + x² = 0

⇒ x²–x–1 = 0