If AM of two numbers be twice their GM then the numbers are in the ratio
anantha srujai , 10 Years ago
Grade 12
Algebra> If AM of two numbers be twice their GM th...
2 Answers
SHAIK AASIF AHAMED
Last Activity: 10 Years ago
Hello student,
Please find the answer to your question below
Let the numbers be a,b
AM=(a+b)/2
GM= [sqrt{ab}]
Given AM=2GM
So (a+b)/2=2 [sqrt{ab}]
a+b-4 [sqrt{ab}] =0
Divide each term by b we get
(a/b)+1-4 [sqrt{a/b}] =0
([sqrt{a/b}] )2-4( [sqrt{a/b}] )+1=0
on solving quadratic equation we get
[sqrt{a/b}] =2 
[sqrt{3}]
So (a/b)=( [2sqrt{3}] )2
(a/b)=7[4sqrt{3}]
Khushi
Last Activity: 6 Years ago
A.T.Q.
a + b / 2 = 2√ab
a + b = 4√ab
Squaring on both sides...
(a + b)² = 16ab ....(i)
a² + b² + 2ab = 16ab
a² + b² + 2ab - 16ab = 0
a² + b² - 14ab = 0
a² + b² - 2ab - 12ab = 0
(a - b)² - 12ab = 0
(a - b)² = 12ab ....(ii)
Dividing (i) by (ii)
(a + b)² / (a - b)² = 16ab / 12ab
(a + b / a - b)² = 4/3
Square root on both sides...
a + b / a - b = 2 /√3
Using componento and dividendo ....
a + b + a - b / a + b - a + b = 2 + √3 / 2 - √3
2a / 2b = 2 + √3 / 2 - √3
a / b = 2 + √3 / 2 - √3
Hence proved.
Hope you can understand it.
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