# The Homogeneous Equation of the Second Degree

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

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

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

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

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