Question

If p, q be two A.M.'s and G be one G.M. between two numbers, then G² = 

Options

• (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) 

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