Class 11

Math

Algebra

Sequences and Series

Sum of two numbers is $6$ times their geometric mean, show that numbers are in the ratio $(3+22 ):(3−22 )$.

Geometric mean of two numbers $a$ and $b$ is $ab $.

According to the question,

$a+b=6ab $

$2ab a+b =13 $

Apply componendo and dividendo.

$a+b−2ab a+b+2ab =3−13+1 $

$(a )_{2}+(b )_{2}−2(a ×b )(a )_{2}+(b )_{2}+2(a ×b ) =24 $

$(a −b a +b )_{2}=12 $

$a −b a +b =12 $

Apply componendo and dividendo again.

$(a −b )−(a −b )(a +b )+(a −b ) =2 −12 +1 $

$a −a +b +b a +a +b −b =2 −12 +1 $

$2b 2a =2 −12 +1 =2 −12 +1 $

$ba =2 −12 +1 $

Square both the sides.

$(ba )_{2}=(2 −12 +1 )_{2}$

$ba =(2 −12 +1 )$

$ba =3−22 3+22 $

Thus, $a$ and $b$ are in the ratio of $3+22 :3−22 $.