If ratio of H.M. and G.M. of two numbers is $12 : 13$ , then prove that ratio of numbers is $4 : 9$ .

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Solution

Let numbers are a and b. then
HM = 2ab/(a+b)
GM = 
HM/GM = 12/13
2 /(a+b) = 12/13
(a+b)/ = 13/6
 +  = 13/6
Let  = x.then
x + 1/x = 13/6
6x² – 13x + 1=0
x = 2/3 or 3/2
a/b = x² = 4/9 or 9/4.

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If ratio of H.M. and G.M. of two numbers is 12:13, then prove that ratio of numbers is 4:9.

Let numbers are a and b. thenHM = 2ab/(a+b)GM = HM/GM = 12/132 /(a+b) = 12/13(a+b)/ = 13/6 + = 13/6Let = x.thenx + 1/x = 13/66x2 – 13x + 1=0x = 2/3 or 3/2a/b = x2 = 4/9 or 9/4.

If ratio of H.M. and G.M. of two numbers is $12 : 13$ , then prove that ratio of numbers is $4 : 9$ .

Let numbers are a and b. then
HM = 2ab/(a+b)
GM = $\sqrt{ab}$
HM/GM = 12/13
2 $\sqrt{ab}$ /(a+b) = 12/13
(a+b)/ $\sqrt{ab}$ = 13/6
$\sqrt{a/b}$ + $\sqrt{b/a}$ = 13/6
Let $\sqrt{a/b}$ = x.then
x + 1/x = 13/6
6x² – 13x + 1=0
x = 2/3 or 3/2
a/b = x² = 4/9 or 9/4.