If p, q be two A.M.'s and G be one G.M. between two numbers, then… - Vidyadip

MCQ

If $p, q$ be two $A.M.'s$ and $G$ be one $G.M.$ between two numbers, then $G^2 =$

✓
$(2\text{p}-\text{q})(\text{p}-2\text{q})$
B
$(2\text{p}-\text{q})(2\text{q}-\text{p})$
C
$(2\text{p}-\text{q})(\text{p}+2\text{q})$
D
None of these.

✓

Answer

Correct option: A.

$(2\text{p}-\text{q})(\text{p}-2\text{q})$

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

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