General Equation of Second Degree
Let A₁x + B₁y + C₁= 0 —————— (i)
and A₂x + B₂y + C₂ = 0 ——————- (ii)
be the equations of two straight lines.
Consider the equation,
(A₁x + B₁y + C₁) (A₂x + B₂y + C₂) = 0 —————— (iii)
The coordinates of any point lying in straight line (1) will satisfy (3) also. Hence all points on the straight line (1) lie on (3). Similarly all points lying on the straight line (2) lie on (3).
Conversely, the coordinates of any point lying on (3) will satisfy (1) or (2) or both. Hence equation (3) represents a pair of the straight lines given by (1) and (2). If the left hand side of equation (3) be expanded we get an equation of second degree in x and y, i.e. an equation of the form,
ax² + 2hxy + by + 2gx + 2f + c=0
Thus we see that the equation of a pair of lines is a second degree equation. But the converse is not always true. As we shall see later, equations of second degree will represent a pair of straight lines only if the left hand side can be resolved into two linear factors.
Homogeneous Equation
An equation with two variables x and y in which each term has the same degree is known as a homogeneous equation. If the degree of each term is 2, then the equation is known as the homogeneous equation of degree two or a second degree homogeneous equation. The most general form of a homogeneous equation of degree two is ax² + 2hxy + by² = 0.
Homogeneous Equation of Second Degree
The Homogeneous Equation of Second Degree ax² + 2hxy + by² = 0, always Represents a Pair of Straight Lines through the Origin.
The homogeneous equation of second degree in x and y is ax² + 2hxy + by² = 0.
The equation can be written as:
y² + (2h/b)xy + ax²/b = 0, if b ≠ 0
