Question

the sum of two numbers is six times their geometric mean show that the number are in the ratio a:b and also find a and b​

Answer 1

Step-by-step explanation:

Let the two numbers are a and b.

Thus their G.M=

ab

According to the given condition, a+b=6

ab

....(1)

⇒(a+b)

2

=36(ab)

Also, (a−b)

2

=(a+b)

2

−4ab=36ab−4ab=32ab

⇒a−b=

32

ab

=4

2

ab

........(2)

Adding (1) and (2) , we obtain 2a=(6+4

2

)

ab

⇒a=(3+2

2

)

ab

Substituting the value of a in (1), we obtain b=6

ab

−(3+2

2

)

ab

⇒b=(3−2

2

)

ab

⇒

b

a

=

(3−2

2

)

ab

(3+2

2)

ab

=

3−2

2

3+2

2

Thus , the required ratio is (3+2

2

):(3−2

2

)