# homogeneous equation

 $f(x,\,y)=0,$

where the left hand 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 of the equation by $y^{r}$.  Then the left 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 of the equation by $y^{2}$ and then solving the equation  $(x/y)^{2}-7(x/y)+10=0$).

• The equation  $2x^{3}-x^{2}y-6xy^{2}+3y^{3}=0$  determines that   $x/y=\frac{1}{2}$  or  $x/y=\pm\sqrt{3}$ (first divide the equation by $y^{3}$ and then solve  $2(x/y)^{3}-(x/y)^{2}-6(x/y)+3=0$).

 Title homogeneous equation Canonical name HomogeneousEquation Date of creation 2013-03-22 15:14:41 Last modified on 2013-03-22 15:14:41 Owner pahio (2872) Last modified by pahio (2872) Numerical id 7 Author pahio (2872) Entry type Topic Classification msc 26C05 Classification msc 26B35 Classification msc 00A99 Related topic Variation Related topic HomogeneousPolynomial Related topic Equation Related topic RegularDecagonInscribedInCircle