The homogeneous equation $$f(x,\,y) = 0,$$ where the left hand side is a homogeneous polynomial of degree $r$ in $x$ and $y$ determines the ratio $x/y$ between the indeterminates. One can be persuaded of this by dividing both sides of the equation by $y^r$ Then the left side depends only on $x/y$ (which may be denoted e.g. by $t$ .

Examples

• The equation $5x+8y = 0$ , determines that $x/y = -\frac{8}{5}$
• The equation $x^2-7xy+10y^2 = 0$ , determines that $x/y = 2$ , or $x/y = 5$ , (these values are obtained by first dividing both sides of the equation by $y^2$ and then solving the equation $(x/y)^2-7(x/y)+10 = 0$ .
• The equation $2x^3-x^2y-6xy^2+3y^3 = 0$ , determines that $x/y = \frac{1}{2}$ , or $x/y = \pm\sqrt{3}$ (first divide the equation by $y^3$ and then solve $2(x/y)^3-(x/y)^2-6(x/y)+3 = 0$ .