Question

If the harmonic mean between two positive numbers is to their geometric mean as 12:13. The numbers are in the ratio

• A. 12:13
  
• B. $\frac{1}{12}:\frac{1}{13}$
  
• C. 4:9
  
• D. $\frac{1}{4}:\frac{1}{9}$

Answer 1

Answer: (C), (D)

The correct options are
C

4:9

D

$\frac{1}{4}:\frac{1}{9}$

By the given condition if the numbers be a and b then
$\frac{2ab}{a+b}:\sqrt{ab}=12:13$
$\frac{2\sqrt{ab}}{a+b}=\frac{12}{13}$
$26\sqrt{ab}=12(a+b).$
Divisible by 2b
$\Rightarrow 13\sqrt{\frac{a}{b}}=6(\frac{a}{b}+1)put\sqrt{\frac{a}{b}}=x6x^2-13x+6=0\therefore (2x-3)(3x-2)=0\Rightarrow x=\frac{3}{2}or\frac{2}{3}=\sqrt{\frac{a}{b}}$
$\therefore \frac{a}{b}=\frac{9}{4}or\frac{4}{9}$
Hence, the numbers are in the ratio 4:9