Question

If a² + b² + c² - ab - bc - ca ≤ 0 then what is the value of (a +b)/c is .

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Answer 1

Answer:

The answer is 2.

Step-by-step explanation:

How?

Well, if you are familiar with the inequality which I'm going to state then you would have got the answer pretty easily. So, here goes.

a2+b2+c2>=ab+bc+aca2+b2+c2>=ab+bc+ac

Did you know this? You probably didn’t. See if you can prove it yourself first.

Okay so, this is one of the proofs:

We know that,

a2+b2>=2aba2+b2>=2ab [By applying AM-GM on a2a2 & b2b2 ]

Similarly, b2+c2>=2bcb2+c2>=2bc & a2+c2>=2aca2+c2>=2ac .

Adding these 3 inequalities, we get, a2+b2+c2>=ab+bc+aca2+b2+c2>=ab+bc+ac [after cancelling the 2]

Well, now its easy to see that, in the above question,

a2+b2+c2=ab+bc+aca2+b2+c2=ab+bc+ac

Now, this can be easily solved by using somewhat the same logic which we use to prove AM-GM. See if you can get the solution.

Okay, here’s what you do:

Multiply the equation by 2 and bring all the terms on the left hand side.

Now, the equation can be written as:

(a−b)2+(b−c)2+(c−a)2=0(a−b)2+(b−c)2+(c−a)2=0

Now, we use the rule, “if sum of some squares is equal to zero then all of them are zero” since a square is always non-negative. (Note that, for this you could have also used the fact that in AM-GM, equality holds when all the terms are equal)

Thus,

a−b=b−c=c−a=0a−b=b−c=c−a=0

So, a=b=ca=b=c .

And,

(a+b)/c=2a/a=2(a+b)/c=2a/a=2 .

There you go, the answer is 2. You’ve probably learned a few important techniques for solving inequalities in this problem, always remember them.

Hope it helps, good luck! =D