SATHEE: Theory of Equations 1 Question 53

Theory of Equations 1 Question 53

54. If $α$ and $β$ are the roots of $x^{2} + p x + q = 0$ and $γ, δ$ are the roots of $x^{2} + r x + s = 0$ , then evaluate $(α - γ) (β - γ)$ $(α - δ) (β - δ)$ in terms of $p, q, r$ and $s$ .

$(1979, 2 M)$

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Answer:

Correct Answer: 54. $((q - s)^{2} - r q p - r s p + s p^{2} + q r^{2})$

Solution:

Since, $α, β$ are the roots of $x^{2} + p x + q = 0$

and $γ, δ$ are the roots of $x^{2} + r x + s = 0$

$\begin{array}{ll} ∴ & α + β = - p, α β = q \\ and & γ + δ = - r, γ δ = s \end{array}$

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

$\begin{aligned} = [α^{2} - (γ + δ) α + γ δ] [β^{2} - (γ + δ) β + γ δ] \\ = (α^{2} + r α + s) (β^{2} + r β + s) \\ = (α β)^{2} + r (α + β) α β + s (α^{2} + β^{2}) + α β r^{2} + r s (α + β) + s^{2} \\ = q^{2} - r q p + s (p^{2} - 2 q) + q r^{2} - r s p + s^{2} \\ = (q - s)^{2} - r q p - r s p + s p^{2} + q r^{2} \end{aligned}$

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Theory of Equations 1 Question 53

54. If α and β are the roots of x2+px+q=0 and γ,δ are the roots of x2+rx+s=0, then evaluate (α−γ)(β−γ) (α−δ)(β−δ) in terms of p,q,r and s.

Answer:

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54. If $α$ and $β$ are the roots of $x^{2} + p x + q = 0$ and $γ, δ$ are the roots of $x^{2} + r x + s = 0$ , then evaluate $(α - γ) (β - γ)$ $(α - δ) (β - δ)$ in terms of $p, q, r$ and $s$ .