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

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The equation $5x+8y=0$ determines that $x/y=-\frac{8}{5}$ .
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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 $(x/y)^{2}-7(x/y)+10=0$ ).
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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