Question

Let G be the geometric mean of two positive numbers a
and b, and M be the arithmetic mean of
1/a and 1/b If M is 4:5, then a: b can be:

(A) 1:4
(B) 1:2
(C) 2:3
(d) 3:4​

Answer 1

Answer:

(A) 1:4

Step-by-step explanation:

Correction : $\frac{1}{M}:G=4:5$

1) We have,

G =Geometric Mean of 'a' and 'b'.

$G=\sqrt{ab}$

2) Also,

M = Arithmetic mean of '1/a' and '1/b'.

$M=\frac{\frac{1}{a}+\frac{1}{b}}{2}= \frac{\frac{a+b}{ab}}{2}=\frac{a+b}{2ab}=>\frac{1}{M}=\frac{2ab}{a+b}$

3)  Now,

$\frac{1}{M}:G=4:5\\ \\=>\frac{\frac{1}{M} }{G}=\frac{4}{5}\\ \\=>\frac{\frac{2ab}{a+b}}{\sqrt{ab} } =\frac{4}{5}\\ \\=>\frac{\sqrt{ab} }{a+b}=\frac{2}{5}\\ \\=>5\sqrt{ab} =2(a+b)\\ \\=>25ab=4(a+b)^2\\ \\=>17ab=4(a^2+b^2)\\ \\=>4a^2-17ab+4b^2=0\\ \\=>4a^2-16ab-ab+4b^2=0\\ \\=>4a(a-4b)-b(a-4b)=0\\ \\=>(a-4b)(4a-b)=0\\ \\=>a:b=1:4 \:or\:4:1$

Hence, Answer can be option (A) 1:4.