Let α,β be the roots of x2−x+p=0 and γ,δ be the roots of x2−4x+q=0. If α,β,γ,δ are in G.P., the integral values of p,q are respectively
Let r be the common ratio of the G.P. α,β,γ,δ then β=αr,γ=αr2 and δ=αr3
∴α+β=1⇒α+αr=1⇒α(1+r)=1......(i)
αβ=p⇒α(αr)=p⇒α2r=p......(ii)
γ+δ=4⇒αr2+αr3=4⇒αr2(1+r)=4.......(iii)
and γδ=q⇒αr2.αr3=q⇒α2r5=q.....(iv)
Dividing (iii) by (i), we get r2=4⇒r=±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).