Question:
If \[ x^2+y^2-xy=3 \] and \[ y-x=1 \] find:
\[ \frac{xy}{x^2+y^2} \]
Solution:
Using identity:
\[ (y-x)^2=x^2+y^2-2xy \]
Substituting the given value:
\[ 1^2=x^2+y^2-2xy \]
\[ 1=x^2+y^2-2xy \]
Given:
\[ x^2+y^2-xy=3 \]
Subtracting the two equations:
\[ (x^2+y^2-xy)-(x^2+y^2-2xy)=3-1 \]
\[ xy=2 \]
Now,
\[ x^2+y^2-xy=3 \]
\[ x^2+y^2-2=3 \]
\[ x^2+y^2=5 \]
Therefore,
\[ \frac{xy}{x^2+y^2} = \frac{2}{5} \]