If a+b=b/a+a/b. Find a sq. +b sq.

Question

^{PRMO questions} 

Answer 1

Hi Pradhuman,

 

Took me a little "playing with" this to figure out how to get a constant instead of a variable.  Here is what I found:

 

a+b = b/a + a/b

 

Get a common denom on right side.

 

a + b = (a² + b²)/ab

 

Mult both sides by ab

 

ab(a+b) = a² + b²

 

Here was the trick to finding a numerical solution....

 

Move both a² and b² to the other side, so that now the right side = 0

 

Also, distribute the ab into (a+b)

 

ab(a+b)-a² - b² = 0

 

a²b + ab² - a² -b² = 0

 

 

Now rearrange so the "a²"s are next to each other and the "b²"s are next to each other so that you can group the first two terms together and so that you can group the last 2 terms together.

 

(a²b - a²) + (ab² - b²) = 0

 

Now factor an a² out of the first group and a b² out of the second group.

 

a²(b-1) + b²(a-1) = 0

 

 

This should look a lot like how to find the zeros of a polynomial.  

 

The numerical answer is a = 1 and b = 1.

 

 Check this.   a+b = 1 + 1 = 2.  

a/b = 1   b/a = 1     a/b + b/a = 1 + 1 = 2.   

 

I think this is the correct answer.