f(x,y) = ax

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

Setting X = (x,y,z) and denoting by X

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

1) J

2) J

3) J

f(x,y) = ax

(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)

To achieve this multiply f by a:

a

Separate the terms involving y:

a

<=> a

Completing the square on the left:

a

<=> (ax + by + d)

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:

y

and the two required lines are (ax + by + d) + (uy + v) = 0 , (ax + by + d) - (uy + v) = 0.

u and v must satisfy u

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.

f(x,y) = ax

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 (J

Lines.html

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ThreeLines.html

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