SATHEE: Theory of Equations 1 Question 55

Theory of Equations 1 Question 55

56. If $α$ and $β$ are the roots of the equation $x^{2} + p x + 1 = 0; γ, δ$ are the roots of $x^{2} + q x + 1 = 0$ , then $q^{2} - p^{2} = (α - γ) (β - γ) (α + δ) (β + δ)$

$(1978, 2 M)$

Show Answer

Answer:

Correct Answer: 56. $(q^{2} - p^{2})$

Solution:

Since, $α + β = - p, α β = 1$ and $γ + δ = - q, γ δ = 1$

Now, $(α - γ) (β - γ) (α + δ) (β + δ)$

$\begin{aligned} = α β - γ (α + β) + γ^{2} α β + δ (α + β) + δ^{2} \\ = 1 - γ (- p) + γ^{2} 1 + δ (- p) + δ^{2} \end{aligned}$

$= (1 + γ^{2} + γ p) (1 - δ p + δ^{2}) = (- q γ + γ p) (- δ p - δ q)$

$[∵ γ^{2} + q γ + 1 = 0 and δ^{2} + q δ + 1 = 0]$

$= (q^{2} - p^{2}) (γ δ) = q^{2} - p^{2}$

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