Question

# 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

Hint:

### We know that A.M is Arithmetic Mean and G.M is Geometric Mean and the A.M and G.M for two positive numbers, say a and b will be and √ respectively. We will make a quadratic equation using it and the roots of the equation gives the value of the numbers which we have to prove.

## The correct answer is: 1:9

### Detailed Solution

AM between two positive numbers is given by

GM between two positive numbers a and b is given by

We have been given that :

We have to be careful while solving quadratic equation questions as there are chances of mistakes with the signs while finding the roots. Sometimes the students try to use factorisation methods to solve the quadratic equation, but in this type of questions, it is not recommended at all. Always try to use the quadratic formula for solving. You can also remember the statement to be proved, given in the question as a property for two positive numbers.

### Related Questions to study

### If where a<1, b<1 then

### If where a<1, b<1 then

### If l, m, n are the terms of a G.P which are +ve, then

### If l, m, n are the terms of a G.P which are +ve, then

### The relationship between force and position is shown in the figure given (in one dimensional case) calculate the work done by the force in displacing a body from x=0 cm to x=5 cm

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### Fifth term of G.P is 2 The product of its first nine terms is

Here note that the fifth term is having fourth power of 2 and not fifth power. We need not to find all nine terms separately; only finding the product is enough because that product will then be written in the form of the term that is known. Terms in a G.P. are having a common ratio in between. That’s why the power of r is increasing as the terms are increasing.

### Fifth term of G.P is 2 The product of its first nine terms is

Here note that the fifth term is having fourth power of 2 and not fifth power. We need not to find all nine terms separately; only finding the product is enough because that product will then be written in the form of the term that is known. Terms in a G.P. are having a common ratio in between. That’s why the power of r is increasing as the terms are increasing.