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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 GP, then the integral values of p and q respectively are

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
The sum of roots and product of roots is follows :
α+β=1,αβ=pγ+δ=4,γδ=q
Let r be the common ratio of the GP α,β,γ,δ
Thenα+β=1andγ+δ=4 α+αr=1andαr2+αr3=4α(1+r)=1andαr2(1+r)=4So(1+r)=1aandar2(1α)=4r=±2
When r = 2, then α=13 which is inadmissible as all given options are integral values.
Hence, r = - 2 is to be considered, then α=1
Thus αβ=p=2and γδ=q=32


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