Grade 12Algebra
If AM of two numbers be twice their GM then the numbers are in the ratio
If AM of two numbers be twice their GM then the numbers are in the ratio
2 Answers
SHAIK AASIF AHAMED
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)=( [2
sqrt{3}] )2(a/b)=7
[4sqrt{3}]Khushi
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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