Question

If p, q be two A.M.'s and G be one G.M. between two numbers, then G² =
(a) (2p − q) (p − 2q)
(b) (2p − q) (2q − p)
(c) (2p − q) (p + 2q)
(d) none of these

Answer 1

(a) (2p − q) (p − 2q)

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{two}\mathrm{numbers}\mathrm{be}a\mathrm{and}b. \\ a,p,q\mathrm{and}b\mathrm{are}\mathrm{in}\mathrm{A}.\mathrm{P}. \\ \therefore p-a=q-p=b-q \\ \Rightarrow p-a=q-p\mathrm{and}q-p=b-q \\ \Rightarrow a=2p-q\mathrm{and}b=2q-p\left(\mathrm{i}\right) \\ \mathrm{Also},a,G\mathrm{and}b\mathrm{are}\mathrm{in}\mathrm{G}.\mathrm{P}. \\ \therefore G^2=ab \\ \Rightarrow G^2=\left(2p-q\right)\left(2q-p\right) \end{gathered}$$