Solving two systems with two unknown?

Question

Let's say if we are giving the following two equations:

$$ 1= X/(X^2 +Y^2) $$ $$ 2= Y/(X^2 +Y^2) $$

How are we going to solve for X and Y [ by HAND ] ?

Why would Summing the squares of the two equations would be one of the approach to solve for X and Y?

Thanks!

Answer 1

If $X$ and $Y$ are not both equal to zero, you can divide one by another. Let's say $X\neq 0$. Then, dividing the second by the first you obtain $Y=2X$. Now plug this into one of the equations.

Answer 2

Note that $$ 5 = 1^2+2^2 = \frac{X^2}{(X^2+Y^2)^2}+\frac{Y^2}{(X^2+Y^2)^2} =\frac{X^2+Y^2}{(X^2+Y^2)^2} = \frac{1}{X^2+Y^2} $$ so $X^2+Y^2=1/5$. The original equations then give $X=1/5$ and $Y=2/5$.