# Minimum value of b

1. Oct 22, 2013

### utkarshakash

1. The problem statement, all variables and given/known data
Let a,b,c be in G.P. and a-b,c-a,b-c in H.P. If both roots of $(a+c)x^2 + bx + 4b^2=0$ are positive and minimum value of 'b' be k then value of |[k]|

2. Relevant equations

3. The attempt at a solution
Let a,b,c be denoted by a,ar,ar^2. now
$\frac{2}{c-a}=\frac{1}{a-b}+\frac{1}{b-c} \\ r^2+4r+1=0$

Since both roots are +ve sum and product of roots should also be +ve.
$\frac{-b}{a+c}>0 \\ \frac{4b^2}{a+c}>0 \\ \\ \frac{r}{1+r^2}<0\\ \frac{4ar}{1+r^2}>0$

2. Oct 22, 2013

### haruspex

I think the left hand term in the last line of your post is wrong.
Can you deduce the sign of a?
It might help to express everything in terms of b and r rather than a and r, since the question concerns b, not a.

3. Oct 23, 2013

### utkarshakash

It should be $\dfrac{4ar^2}{1+r^2}$. Now if I change a to b/r it changes into$\dfrac{4br}{1+r^2}>0$. Now denominator will always be +ve so I can simply write br>0. Also from the first inequality I can say that r<0. To satisfy both inequalities b must be less than zero. Now b=ar<0. Thus a must be +ve.

4. Oct 23, 2013

### haruspex

The info in the OP implies both roots are real, since a complex root cannot be said to be positive or negative. You haven't used that.

5. Oct 23, 2013

### utkarshakash

b(1+r)<1/16

6. Oct 23, 2013

### haruspex

I get something a little different. Pls post your working.

7. Oct 23, 2013

### utkarshakash

Sorry. I made a mistake. It should be $\dfrac{16br^2-r+16b}{r} \leq 0$. I have used the fact that discriminant of the original equation is greater than 0 and substituting a as b/r. As r<0 (deduced earlier) $16br^2-r+16b \geq 0$. This means that discriminant of the equation in LHS is <=0. $b^2 \geq \frac{1}{256*4}$. From here minimum value of b is 1/32. Thus |[1/32]|=0. But this is not the answer.

8. Oct 24, 2013

### haruspex

You already determined that r + 1/r = -4, so use that to eliminate r.

9. Oct 24, 2013

### utkarshakash

OK. This gives me the correct answer. But I want to know what was wrong in my solution?

10. Oct 24, 2013

### haruspex

Because you did not use all the available information you did not get a strong enough bound. Note that even now you have not proved your answer to be correct. To do that you would need to demonstrate that every b > k occurs in a solution of the given equations.

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