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Question

Let α,β be the roots of x2x+p=0 and γ,δ be the roots of x24x+q=0. If α,β,γ,δ are in G.P., the integral values of p,q are respectively

A
– 2, – 32
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B
– 2, 3
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C
– 6, 3
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D
– 6, – 32
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Solution

The correct option is A – 2, – 32

Let r be the common ratio of the G.P. α,β,γ,δ then β=αr,γ=αr2 and δ=αr3
α+β=1α+αr=1α(1+r)=1......(i)
αβ=pα(αr)=pα2r=p......(ii)
γ+δ=4αr2+αr3=4αr2(1+r)=4.......(iii)
and γδ=qαr2.αr3=qα2r5=q.....(iv)
Dividing (iii) by (i), we get r2=4r=±2
If we take r=2, then α is not integral, and thus p and q will also be not integral, so we take r=2,
Substituting r=2 in (i), we get α=1
Now, form (ii), we have p=α2r=(1)2(2)=2
and from (iv), we have q=α2r5=(1)2(2)5=32
(p,q)=(2,32).


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