homogeneous equation

The 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  $2x^3-x^{2}y-6x{y}^2+3y^3=0$  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