Question

If $A$ and $G$ are A.M. and G.M. between two positive numbers $a$ and $b$ are connected by the relation A+G=a-b then the numbers are in the ratio

• A. $1:3$
• B. $1:6$
• C. $1:9$
• D. $1:12$

Answer 1

The correct option is C $1:9$

AM between two positive numbers is given by

$\frac{a+b}{2}$

GM between two positive numbers a and b is given by

$\sqrt{ab}$

Given that

$A+G=a-b$

$⟹\frac{a+b}{2}+\sqrt{ab}=a-b$

$⟹a+b+2\sqrt{ab}=2a-2b$

$⟹a-3b-2\sqrt{ab}=0$

Dividing the above equation by b

$\frac{a}{b}-3-2\sqrt{\frac{a}{b}}=0$

Assuming $\sqrt{\frac{a}{b}}=t$

The equation becomes

$t^2-2t-3=0⟹t^2-3t+t-3=0⟹(t-3)(t+1)=0⟹t=3,-1$

If $t=3$

then $\sqrt{\frac{a}{b}}=3$

or $a:b=9:1$