## 1. Degenerate conics

Conics are described in cartesian coordinates through equations:
f(x,y) = ax2 + 2bxy + cy2 + 2dx + 2ey + f = 0.
Degenerate conics are those for which the determinant of the corresponding symmetric matrix M is zero:

The matrix serves to represent the conic as a quadratic form in homogeneous coordinates.
Setting  X = (x,y,z) and denoting by Xt the transposed column-vector this is:

The equation f(x,y) =0 results by setting z=1: f(x,y) = F(x,y,1) =0. By the way, this representation justifies the use of "twos" in the first formula for f(x,y).
If f(x,y) is a product of lines  Ax+By+C, A'x+B'y+C', then it is easily seen that the resulting matrix M has determinant 0.
In fact M has then the form:

The nature of the conic is determined by its three invariants: J1=a+c, J2=ac-b2 and J3=|M|, last denoting the determinant of the matrix. This is true for every conic. In particular for degenerate ones we have:
1)  J3=0 and J2 >0, then  two imaginary lines                   (example: x2+y2=0).
2)  J3=0 and J2 <0, then  two real intersecting lines         (example: x2-y2=0).
3)  J3=0 and J2 = 0, then two real parallel lines               (example: x2+x=0).

## 2. Determining the factors

Assuming the degenerate conic in the general form:
f(x,y) = ax2 + 2bxy + cy2 + 2dx + 2ey + f = 0,
(with |M|=0) the determination of the two lines from the coefficients involves a simple calculation,
which represents f(x,y) as a difference of squares  f(x,y) = m(x,y)2 - n(x,y)2, the two lines being then m(x,y)+n(x,y) and m(x,y)-n(x,y) (see Loney p. 95).
To achieve this multiply f by a:
Separate the terms involving y:
<=>   a2x2+2ax(by+d)      = -(acy2+2aey+af).
Completing the square on the left:
a2x2+2ax(by+d)+(by+d)2  = (by+d)2-(acy2+2aey+af),
<=> (ax + by + d)2  =  y2[b2-ac]+2y[bd-ae]+[d2-af].
The condition |M|=0 is easily seen to be equivalent to the vanishing of the discriminant of the right-side quadratic polynomial in y. In this case the right side can be written in the form:
y2[b2-ac]+2y[bd-ae]+[d2-af] = (uy+v)2,
and the two required lines are                                                 (ax + by + d) + (uy + v) = 0 ,    (ax + by + d) - (uy + v) = 0.
u and v must satisfy  u2 = b2-ac,   v2 = d2-af   and   uv = bd-ae, later allowing the right selection of signs for u, v in terms of the given coefficients.

## 3. Example

As an example (Loney p.97) resolve   12x2+7xy-10y2+13x+45y-35 = 0  in two linear factors.
Doing the computations of the previous section it turns out that  |u|=23/2, |v| = 43/2 and  uv<0.
Thus the signs must be opposite, giving   u=23/2  and  v=-43/2  and consequently:
(ax+by+d)+(uy+v) = (12x+(7/2)y+(13/2)) + ((23/2)y - (43/2)) = 0,
giving the factor           4x + 5y - 5 = 0.
Analogously                  (ax+by+d)-(uy+v) = (12x+(7/2)y+(13/2)) - ((23/2)y - (43/2)) = 0,
giving the factor           3x - 7y + 7 = 0.

## 4. Varying the constant

Starting with a degenerate conic and varying the constant f we obtain in general non-degenerate conics:
f(x,y) = ax2 + 2bxy + cy2 + 2dx + 2ey + f = k.
These are level-curves of the function f(x,y) and if the degenerate equation resolves to two intersecting real lines then the corresponding level surfaces are hyperbolas having these lines as asymptotes.

This results immediately from the fact that an equation of the form  (ax+by+c)(a'x+by'+c') = k (k non-zero) represents a hyperbola with respect to its asymptotes which are the two lines on the left side (see HyperbolaAsymptotics.html ).

There is a similar behavior for the other cases of degenerate conics. For instance in the case the degnerate conic represents two parallel lines (J2=0 in section 1) the level curves are again imaginary or real parallel lines. For example if f(x,y)=x2+x, depending on the discriminant of  x2+x-k, the equation f(x,y)=k represents two imaginary or two real parallel lines. Similarly in the case of J2>0 as is for f(x,y) = x2+y2, setting f(x,y) = k we obtain real or imaginary circles.

HyperbolaAsymptotics.html
Lines.html
LinePencils.html
ThreeLines.html

### Bibliography

Loney, S. L. Coordinate Geometry. Delhi, AITBS Publishers, 1991, p. 97.