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Let the harmonic mean and geometric mean...

Let the harmonic mean and geometric mean of two positive numbers be in the ratio 4:5. Then the two numbers are in ratio............ (1992, 2M)

Text Solution

Verified by Experts

The correct Answer is:
`4 : 1`
Let the two positive number be ka and `a, a gt 0`
Then, `G = sqrt(ka.a) = sqrtk.a`
and `H = (2(ka) a)/(ka + a) = (2ka)/(k +1)`
Again, `(H)/(G) = (4)/(5)` [given]
`rArr ((2ka)/(k+1))/(sqrtk a) = (4)/(5) rArr (2 sqrtk)/(k +1) = (4)/(5)`
`rArr 5 sqrtk = 2k + 2`
`rArr 2k - 5 sqrtk + 2 = 0`
`rArr sqrtk = (5 +- sqrt(25 -16))/(4) = (5 +- 3)/(4) = 2, (1)/(2)`
`rArr k = 4, 1//4`
Hence, the required ratio is `4 : 1`
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