homogeneous equationHomogeneousEquation2005-05-07 14:29:412006-02-25 04:53:36Topichomogeneous polynomialproportional% this is the default PlanetMath preamble. as your knowledge
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\newtheorem*{thmplain}{Theorem}The {\em homogeneous equation}
$$f(x,\,y) = 0,$$
where the left hand \PMlinkescapetext{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 \PMlinkescapetext{sides} of the equation by $y^r$.\, Then the left \PMlinkescapetext{side} depends only on $x/y$ (which may be denoted e.g. by $t$).
\textbf{Examples}
\begin{itemize}
\item The equation\, $5x+8y = 0$\, determines that\, $x/y = -\frac{8}{5}$.
\item 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 \PMlinkescapetext{sides} of the equation by $y^2$ and then solving the equation\, $(x/y)^2-7(x/y)+10 = 0$).
\item 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\, $2(x/y)^3-(x/y)^2-6(x/y)+3 = 0$).
\end{itemize}