Question

If

a.m and g.m of two numbers are 13 and 12 respectively.find the numbers

Answer 1

I hope this answer helps you

Answer 2

$Let \: a \:and \: b \: are \: two \: numbers .$

$A.M \: of \: of \: a \: and \:b = 13 \: (given)$

$\implies \frac{a+b}{2} = 13$

$\implies a + b = 26 \: ---(1)$

$G.M \: of \: of \: a \: and \:b = 12\: (given)$

$\implies \sqrt{ab}= 12$

$\implies ab = 12^{2}\\\implies ab = 144 \: ---(2)$

$(a-b)^{2} = (a+b)^{2} - 4ab \\= 26^{2} - 4 \times 144\\= 676 - 576\\= 100$

$\implies a - b = \pm \sqrt{10^{2}} \\= \pm 10\: --(3)$

Case 1:

$If \: a - b = 10, \: and \: a + b = 26$

/* add these two equations , we get */

$2a = 36$

$\implies a = \frac{36}{2}$

$\implies a = 18$

/* Put a = 18 in a + b = 26 */

$b = 26 - 18 = 8$

Case 2:

$If \: a - b = -10, \: and \: a + b = 26$

/* add these two equations , we get */

$2a = 16$

$\implies a = \frac{16}{2}$

$\implies a = 8$

/* Put a = 8 in a + b = 26 */

$b = 26 - 8 = 18$

Therefore.,

$\red { Required \:two \: numbers } \\\green {= (18,8) \:Or \: (8,18) }$

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