Question

If $A_1,A_2;G_1,G_{2}and{H}_1,H_2$ be two A.M.s, G.M.s and H.M.s between two quantities '$a$' and '$b$' , then $A_1{H}_2=A_2{H}_1=G_1{G}_2$= ?

• A. $ab$
• B. $a$
• C. $b$
• D. none of these

Answer 1

The correct option is A $ab$

$a,A_1,A_2,b$ are in A.P.

$a,H_1,H_2,b$ are in H.P.

$\therefore \frac{1}{a},\frac{1}{H_1},\frac{1}{H_2},\frac{1}{b}$ are in A.P.

Multiply by ab.

$\therefore b,\frac{ab}{H_1},\frac{ab}{H_2},a$ are in A.P.

Take in reverse order.

or $a,\frac{ab}{H_2},\frac{ab}{H_1},b$ are in A.P.

Now,

$\therefore A_1=\frac{ab}{H_2}and{A}_2=\frac{ab}{H_1}$

$\therefore A_1{H}_2=A_2{H}_1=ab=G_1{G}_2$