To solve the given problem, let's start by defining the expressions for [tex]\(a\)[/tex] and [tex]\(b\)[/tex] as given in the question:
[tex]\[ a = \frac{x}{x+y} \][/tex]
[tex]\[ b = \frac{y}{x-y} \][/tex]
We are asked to find the value of the expression [tex]\(\frac{a b}{a + b}\)[/tex].
First, we find the product [tex]\(a \cdot b\)[/tex]:
[tex]\[
a \cdot b = \left( \frac{x}{x+y} \right) \cdot \left( \frac{y}{x-y} \right) = \frac{x \cdot y}{(x+y)(x-y)}
\][/tex]
Next, we find the sum [tex]\(a + b\)[/tex]:
[tex]\[
a + b = \frac{x}{x+y} + \frac{y}{x-y}
\][/tex]
To add these fractions, we need a common denominator. The common denominator is [tex]\((x+y)(x-y)\)[/tex]. Let's rewrite both fractions with this common denominator:
[tex]\[
a = \frac{x}{x+y} = \frac{x(x-y)}{(x+y)(x-y)} = \frac{x^2 - xy}{(x+y)(x-y)}
\][/tex]
[tex]\[
b = \frac{y}{x-y} = \frac{y(x+y)}{(x-y)(x+y)} = \frac{xy + y^2}{(x+y)(x-y)}
\][/tex]
Adding these fractions, we get:
[tex]\[
a + b = \frac{x^2 - xy}{(x+y)(x-y)} + \frac{xy + y^2}{(x+y)(x-y)} = \frac{(x^2 - xy) + (xy + y^2)}{(x+y)(x-y)}
\][/tex]
Simplify the numerator:
[tex]\[
(x^2 - xy) + (xy + y^2) = x^2 - xy + xy + y^2 = x^2 + y^2
\][/tex]
So,
[tex]\[
a + b = \frac{x^2 + y^2}{(x+y)(x-y)}
\][/tex]
Now we have the expressions for [tex]\(a \cdot b\)[/tex] and [tex]\(a + b\)[/tex]:
[tex]\[
a \cdot b = \frac{xy}{(x+y)(x-y)}
\][/tex]
[tex]\[
a + b = \frac{x^2 + y^2}{(x+y)(x-y)}
\][/tex]
We need to find:
[tex]\[
\frac{a b}{a + b} = \frac{\frac{xy}{(x+y)(x-y)}}{\frac{x^2 + y^2}{(x+y)(x-y)}} = \frac{xy}{x^2 + y^2}
\][/tex]
Thus, the value of [tex]\(\frac{a b}{a + b}\)[/tex] is:
[tex]\[
\boxed{\frac{xy}{x^2 + y^2}}
\][/tex]
