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If A and G are A.M. and G.M. between two positive numbers a an

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If $A$ and $G$ are A.M. and G.M. between two positive numbers $a$ and $b$ are connected by the relation A+G=a-b then the numbers are in the ratio

1 : 3

1 : 6

1 : 9

1 : 12

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Noah Discussed

A

and

G

are A.M. and G.M. between two positive numbers

a

and

b

are connected by the relation A+G=a-b then the numbers are in the ratio

7 mins ago

Discuss this question LIVE

7 mins ago

Text solutionVerified

AM between two positive numbers is given by

\frac{a + b}{2}

GM between two positive numbers a and b is given by

ab

Given that

A + G = a - b

⟹ \frac{a + b}{2} + ab = a - b

⟹ a + b + 2 ab = 2 a - 2 b

⟹ a - 3 b - 2 ab = 0

Dividing the above equation by b

\frac{a}{b} - 3 - 2 \frac{a}{b} = 0

Assuming

\frac{a}{b} = t

The equation becomes

t^{2} - 2 t - 3 = 0 ⟹ t^{2} - 3 t + t - 3 = 0 ⟹ (t - 3) (t + 1) = 0 ⟹ t = 3, - 1

t = 3

then

\frac{a}{b} = 3

a : b = 9 : 1

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Question Text	If $A$ and $G$ are A.M. and G.M. between two positive numbers $a$ and $b$ are connected by the relation A+G=a-b then the numbers are in the ratio
Answer Type	Text solution:1
Upvotes	150