Question
Question: If α, β are the roots of equation *x*<sup>2</sup> + p*x* + q = 0 and γ, δ are roots of equation *x*<...
If α, β are the roots of equation x2 + px + q = 0 and γ, δ are roots of equation x2 + rx + s = 0 then the value of (α – γ)2 + (β – γ)2 +(α – δ)2 + (β – δ)2 is –
A
2(p2 + r2 – pr + 2q – 2s)
B
2(p2 + r2 – pr + 2q + 2s)
C
2(p2 + r2 – pr – 2q – 2s)
D
None of these
Answer
2(p2 + r2 – pr – 2q – 2s)
Explanation
Solution
α + β = – p , αβ = q
γ + δ = –r, γδ = s
Given expression
= 2(α2 + β2) + 2(γ2 + δ2) – 2 (αγ + βγ + αδ + βδ)
= 2{(α + β)2 – 2αβ} + 2 {(γ + δ)2 – 2 γδ} –2 (α + β) (γ + δ)
= 2(p2 – 2a) + 2(r2 – 2s) – 2(–p) (–r)
= 2 (p2 + r2 – pr – 2q – 2s).