The sum and difference of the arithmetic mean and the geometric mean of two positive integers are re

Question

The sum and difference of the arithmetic mean and the geometric mean of two positive integers are respectively, 18 and 8 . Then the values of the two numbers are\

Professor PC Thomas Classes · Answer

(e) : We have, A.M. + G.M. = 18   .......(i)\ A.M. - G.M. $=8$   .......(ii)\ From (i) and (ii), we get 2 A.M. $=26$\ $\Rightarrow \quad$ A.M. $=13 \Rightarrow \frac{a+b}{2}=13 \Rightarrow a+b=26$      .......(iii)\  From (i), G.M. = 5\ $\Rightarrow \sqrt{a b}=5 \Rightarrow a b=25$\ $\Rightarrow a(26-a)=25$\ [From (iii)]\ $\Rightarrow \quad 26 a-a^{2}=25 \Rightarrow a^{2}-26 a+25=0$\ $\Rightarrow \quad a^{2}-25 a-a+25=0 \Rightarrow(a-25)(a-1)=0$\ Either $a=25$ or $a=1$\ From (iii), we get $b=1$ or $b=25$\ Hence, values of two numbers are 1 and 25.\