Question

If a+b : √ab = 2 : 1,find the value of a : b

Answer 1

as  a+b=2rootab
squaring bth side..  (a+b)^2=4ab
expanding terms,  a^2+b^2+2ab-4ab=0
or  (a-b)^2=0
hence  a=b
so a/b=1

Answer 2

$\frac{a+b}{ \sqrt{ab} } = \frac{2}{1}$
or, $\frac{a}{ \sqrt{ab} } + \frac{b}{ \sqrt{ab} } =2$
or, $\sqrt{ \frac{a}{b} } + \sqrt{ \frac{b}{a} } =2$
LET, $X = \sqrt{ \frac{a}{b} } ,then, \frac{1}{X} = \sqrt{ \frac{b}{a} }$
now,  $X+ \frac{1}{X} =2$
or, $X^{2} -2x+1=0$
or, $( X-1)^{2} =0$
therefore X = 1
so, $\sqrt{ \frac{a}{b} } =1$
or, a/b =1