Homogeneous Equation/Examples/Arbitrary Example 2

Example of Homogeneous Equation

$x^2 + y^2 = 3$

We note that $x^2 + y^2$ is a homogeneous expression:

$\ds \map E {x, y}$

$=$

$\ds x^2 + y^2$

$\ds \leadsto \ \ $

$\ds \map E {k x, k y}$

$=$

$\ds k^2 x^2 + k^2 y^2$

$\ds $

$=$

$\ds k^2 \paren {x^2 + y^2}$

$\ds $

$=$

$\ds k^2 \map E {x, y}$

However, by definition of homogeneous equation, it has to be equated to $0$.

We further observe that $k^2 \paren {x^2 + y^2} = 3 k^2$ not $3$, showing that the complete equation as an expression is not homogeneous.

$\blacksquare$