Suppose that x and y are positive real numbers satisfying x^2 +y^2 =4xy . Then x−y/x+y can be written as �ãa/b, where a and b are coprime positive integers. Find a+b .
To find the value of x - y / x + y, we can first simplify the expression using algebraic manipulation.
Given that x^2 + y^2 = 4xy, we can rewrite the equation as:
(x - y)^2 = 2xy
Expanding the square, we have:
x^2 - 2xy + y^2 = 2xy
Rearranging the terms, we get:
x^2 - 4xy + y^2 = 0
We can now use this equation to find the value of x - y / x + y.
Dividing both sides of the equation by xy, we have:
(x^2 - 4xy + y^2) / xy = 0 / xy
Simplifying the left side of the equation, we get:
(x - 2y)^2 / (xy) = 0
Since the expression on the left is equal to zero, we know that the numerator must be zero:
(x - 2y)^2 = 0
Taking the square root of both sides, we have:
x - 2y = 0
Solving for x, we get:
x = 2y
Now, substitute this value of x into the expression x - y / x + y:
(2y - y) / (2y + y) = y / 3y = 1/3
Therefore, we have found that x - y / x + y can be written as 1/3.
Since the numerator and denominator (1 and 3) are coprime positive integers, the values of a and b are 1 and 3 respectively.
So, a + b = 1 + 3 = 4.
Hence, the value of a+b is 4.