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Find two positive numbers whose Arithmetic Means increased by 2 than Geometric Means and their difference is 12.

Asked by Topperlearning User 5th October 2016, 7:27 AM
Answered by Expert
Answer:
Let the two numbers be m and n. Then,
m-n=12 ........................ (i)
It is given that AM-GM=2
⇒ (m + n)/2-√mn = 2
⇒ (m + n)-2√mn = 4
⇒ (√m-√n)2 = 4
⇒ √m-√n = ±2 ........................ (ii)
Now, m-n= 12
⇒ (√m + √n)(√m-√n) = 12
⇒ (√m + √n)(±2) = 12 ........................ (iii)
⇒ √m + √n = ± 6, [using (ii)]
Solving (ii) and (iii), we get m = 16, n = 4
Hence, the required numbers are 16 and 4.
Answered by Expert 5th October 2016, 9:27 AM
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