Question

If two equations are ${x^2} + {y^2} - xy = 3$ and $y - x = 1$ then find $\dfrac{{xy}}{{{x^2} + {y^2}}}$.

Answer 1

Hint - Use the equation $y - x = 1$, square both sides of the equation and then use the other equation given to solve the question.

Complete step-by-step answer:
Given ${x^2} + {y^2} - xy = 3 \to (1)$ and $y - x = 1 \to (2)$
Now squaring both sides on the equation, $y - x = 1$
We get-
$
  {(y - x)^2} = {1^2} \\
   \Rightarrow {y^2} + {x^2} - 2xy = 1 \\
   \Rightarrow {x^2} + {y^2} - 2xy = 1 - (3) \\
$
Subtracting equation (1) from equation (3)-
$
  ({x^2} + {y^2} - 2xy) - ({x^2} + {y^2} - xy) = 1 - 3 \\
   \Rightarrow - 2xy + xy = - 2 \\
   \Rightarrow - xy = - 2 \\
   \Rightarrow xy = 2 \\
$
Now putting the value of xy in equation (1), we get-
$
  {x^2} + {y^2} - 2 = 3 \\
  {x^2} + {y^2} = 5 \\
 $
Now putting the values of $xy = 2$ and ${x^2} + {y^2} = 5$ in $\dfrac{{xy}}{{{x^2} + {y^2}}}$ , we get-
$\dfrac{{xy}}{{{x^2} + {y^2}}} = \dfrac{2}{5}$
Therefore, the value of $\dfrac{{xy}}{{{x^2} + {y^2}}}$ is 2/5.

Note- Whenever such types of questions appear, then write down the equation given in the question. By squaring both sides on equation (2) and then subtracting equation (1) from the equation (3), we find the value of xy and then putting the value of xy we get ${x^2} + {y^2} = 5$ . And then substituting the values of $xy = 2$ and ${x^2} + {y^2} = 5$ to find the value of $\dfrac{{xy}}{{{x^2} + {y^2}}}$ .