Question

If A and G are AM and GM between two positive numbers a and b are connected by the relation A + G =a - b then the numbers are in the ratio of ????
Please help fast
Will choose brainliest

Answer 1

The numbers are in the ratio of $a:b=9:1$

Step-by-step explanation:

AM between two positive numbers is given by

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

GM between two positive numbers a and b is given by

$G=\sqrt{ab}$

Given that

$A+G=a-b$

$\implies \frac{a+b}{2}+\sqrt{ab}=a-b$

$\implies a+b+2\sqrt{ab}=2a-2b$

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

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

$t^2-2t-3=0$

$\implies t^2-3t+t-3=0$

$\implies t(t-3)+1(t-3)=0$

$\implies (t-3)(t+1)=0$

$\implies t=3, -1$

If $t=3$

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

or, $\frac{a}{b}=9$

If $t=-1$

Then $\sqrt{\frac{a}{b}}=-1$

or, $\frac{a}{b}=1$

Therefore, in this case $a=b$

The AM will be $A=\frac{a+a}{2}=a$

And GM will be $G=\sqrt{a\times a}=a$

$A+G=a+a=2a$

and $a-b=a-a=0$

Hence LHS and RHS are not equal if $a=b$

Therefore,

$\frac{a}{b}=9$

or, $a:b=9:1$

Hope this helps.