The Homogeneous Equation of the Second Degree

General Equation of the Second Degree
The equation of the form is

a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0

When a, b and h are not simultaneously zero, is called the general equation of the second degree or the quadratic equation in x and y.

Homogeneous Equation
An equation of the form f\left( {x,y} \right) = 0 is said to be the homogeneous equation of degree n, where n is a positive integer, and if for some real number k, we have

f\left( {kx,ky} \right) = {k^n}f\left( {x,y} \right)

For example, the equation f\left( {x,y} \right) = {x^4} - 3{x^3}y + 9{x^2}{y^2} + 8{y^4} is a homogeneous equation of degree 4, because

\begin{gathered} f\left( {kx,ky} \right) = {\left( {kx} \right)^2} - 3{\left( {kx} \right)^3}\left( {ky} \right) + 9{\left( {kx} \right)^2}{\left( {ky} \right)^2} + 8{\left( {ky} \right)^4} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {k^4}{x^4} - 3{k^4}{x^3}y + 9{k^4}{x^2}{y^2} + 8{k^4}{y^4} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {k^4}\left[ {{x^4} - 3{x^3}y + 9{x^2}{y^2} + 8{y^4}} \right] \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {k^4}f\left( {x,y} \right) \\ \end{gathered}

But the general second degree equation
f\left( {x,y} \right) = a{x^2} + 2kxy + b{y^2} + 2gx + 2fy + c is not homogeneous equation, because

\begin{gathered} f\left( {kx,ky} \right) = a{\left( {kx} \right)^2} + 2h\left( {kx} \right)\left( {ky} \right) + b{\left( {ky} \right)^2} + 2g\left( {kx} \right) + 2f\left( {ky} \right) + c \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = a{k^2}{x^2} + 2h{k^2}xy + b{k^2}{y^2} + 2gkx + 2fky + c \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {k^2}\left[ {a{x^2} + 2hxy + b{y^2} + \frac{{2g}}{k}x + \frac{{2f}}{k}y + c} \right] \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \ne {k^2}\left[ {a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c} \right] \\ \end{gathered}

Second Degree Homogeneous Equation
The equation of the form a{x^2} + 2hxy + b{y^2} = 0 is called the second degree homogeneous equation.