Question

the HM between 2 numbers is 16/5, their AM is A and GM is G. if 2A+G^2=26, then the numbers are?​

Answer 1

Answer:

$a\;=\;8, b\;=\;2$

Step-by-step explanation:

Given H.M of a and b is $\frac{2ab}{a+b}\;=\;\frac{16}{5}$

$a+b\;=\;\frac{5ab}{8}\;\rightarrow(1)$

Given $2A+G^2\;=\;26$

$2(\frac{a+b}{2})+ab\;=\;26$

$(a+b)+ab\;=\;26$

$\frac{5ab}{8}+ab\;=\;26$

$ab\;=\;16$

From (1),

$a+b\;=\;\frac{5}{8}(16)$

$a+b\;=\;10\;\rightarrow(2)$

∴ $(a-b)^2\;=\;(a=b)^2-4ab\;=100-64\;=36$

$(a-b)\;=\;6\;\rightarrow(3)$

Solve (2) and (3),

∴ $a\;=\;8, b\;=\;2$