<s>[INST] Find the value of $x +y$ If $a=\frac{x}{x^2+y^2}$ and $b=\frac{y}{x^2+y^2}$ then find $x+y$
I find that $x+y/y=\frac{a+b}{b}$ but the ans in the form of a and B only.
 [/INST] Note that:
$$ax+by=1 \qquad \qquad (1)\\
\frac ax=\frac by=\frac{a+b}{x+y} \qquad \qquad (2)$$
From $(2)$ we also get:
$$x=\frac{a(x+y)}{a+b}; y=\frac{b(x+y)}{a+b} \qquad \qquad (3) $$
Plug $(3)$ into $(1)$:
$$\frac{a^2(x+y)}{a+b}+\frac{b^2(x+y)}{a+b}=1 \Rightarrow x+y=\frac{a+b}{a^2+b^2}.$$
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