Solve xy/x^2+y^2=c

$$\frac { x y } { x ^ { 2 } + y ^ { 2 } } = c$$

Solve for c

$$c=\frac{xy}{x^{2}+y^{2}}$$

$$y\neq 0\text{ or }x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x^{2}+y^{2}.

$$xy=c\left(x^{2}+y^{2}\right)$$

Use the distributive property to multiply c by x^{2}+y^{2}.

$$xy=cx^{2}+cy^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$cx^{2}+cy^{2}=xy$$

Combine all terms containing c.

$$\left(x^{2}+y^{2}\right)c=xy$$

Divide both sides by x^{2}+y^{2}.

$$\frac{\left(x^{2}+y^{2}\right)c}{x^{2}+y^{2}}=\frac{xy}{x^{2}+y^{2}}$$

Dividing by x^{2}+y^{2} undoes the multiplication by x^{2}+y^{2}.

$$c=\frac{xy}{x^{2}+y^{2}}$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{y\left(\sqrt{1-4c^{2}}-1\right)}{2c}\text{; }x=\frac{y\left(\sqrt{1-4c^{2}}+1\right)}{2c}\text{, }&c\neq 0\text{ and }y\neq 0\text{ and }|c|\leq \frac{1}{2}\\x=0\text{, }&c=0\text{ and }y\neq 0\\x\neq 0\text{, }&c=0\text{ and }y=0\end{matrix}\right.$$