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When a plane intersects the surface of a cone, curves are formed by the points of intersection called conic sections or conics. Second-degree polynomial equations define the curves. Based on the equation, the conics are degenerate conics or non-degenerate conics. The types of curves formed to give the three types of conics, the hyperbola, the parabola, and the ellipse, which can be either degenerate conics or non-degenerate conics.
Degenerate Conics
If the equation of the curves (conic sections) can be expressed as a product of two polynomials, then they are called degenerate conics. Degenerate conics have a reducible polynomial equation. When the plane intersects a double cone, the conic section obtained is a degenerate conics and forms different types of degenerate conics.
Types of Degenerate Conics
In the real plane, the degenerate conics formed by the intersection planes are of few types. Depending on the plane’s orientation intersecting the double cones, degenerate conics form two lines that may be crossing or parallel, one single line (formed by two coinciding lines) or one double line. These are reducible.
Also, degenerate conics are formed by a lack of enough points to form a curve, such as a single point or no point in the real plane.
Examples of Degenerate Cones
In x2 – y2 = 0 the conic section with the equation is an example of degenerate cones as it can be reduced to (x+y)(x-y) = 0 . It forms two lines that intersect each other.
The equations derived from ax2 + by2 = c for either a or b equals 0 are examples of degenerate cones.
Another example of degenerate cones is the curve with equation x2 + y2 – 2x + 1 = 0.
Non-Degenerate Conics
Conic sections that have equations that are not reducible are called non-degenerate conics. They are curves that can be represented by the general equation A1x2 + A2xy + A3y2 + A4x + A5y + A6 = 0, where A1, A2, A3, A4 , A5 and A6, are constants and A1, A2, and A3 are not equal to zero together.
Based on the constants of the equation, we get the conic sections, the ellipse, the parabola, and the hyperbola.
Ellipse
The curve is an ellipse when A1 and A3 are not equal to 0 and A1A3 > 0 (when A1=A2, the curve is a circle, which is a special case of an ellipse) such that A22 – 4 A1A3 < 0.
E.g., 8x2 – 2xy + 3y2 + 3y – 13 = 0
x2 + xy + 2y2+13x + y = 0 represents an ellipse.
Parabola
The curve is a parabola when either A1 or A3 is zero and A22 – 4 A1A3 = 0.
E.g., 3x2 – 5x + 3y = 0 represents a parabola.
Hyperbola
The curve is a hyperbola when A1A3<0 and A22 – 4 A1A3 > 0.
E.g., 15x2 – 12y2 – 6 (x-y) = 0
-4x2 + y2 +2x+8y+11= 0 represents a hyperbola.
Classifications of Conics as Degenerate or Non-Degenerate
A conic section with the general equation A1x2 + A2xy + A3y2 + A4x + A5y + A6 = 0 can be classified as degenerate conics or non-degenerate conic by the discriminant of its homogenised quadratic form in (x,y,z) which is A1x2 + A2xy + A3y2 + A4xz + A5yz + A6z2 = 0.
The discriminant is the matrix,
A1 A2 A4
Q = A2 A3 A5
A4 A5 A6
The conic is degenerate if the determinant of Q is 0.
Example: Let the equation of a conic be x2+y2+4x+2y+5=0.
Here A1 = 1, A2 = 0, A3 = 1, A4 = 4 , A5 = 2 and A6 = 5.
Therefore,
1 0 1
Q = 0 1 2
4 2 5
Here, the determinant of Q = 4A1A3A6−A1A52-A22A6+A2A4A5−A3A42 = 0. Hence, it is a degenerate conic.
Conclusion
Conic sections are an essential family of curves consisting of parabolas, hyperbolas, and ellipses. It is formed as a plane intersects the surface of a cone or double cone. The conics become degenerate if certain conditions are met. Such as cutting the double cone in the vertex, etc. Even a family of curves containing non-degenerate conics can degenerate in exceptional cases (as seen in the examples of degenerate cones). Hence understanding degenerate conics helps in understanding the limitations of non-degenerate conics.
A general equation can express all the conics. The conic can be determined as an ellipse, parabola, or hyperbola by examining the coefficients of the general equation. The equation also says if the conic is degenerate. It can be identified by the determinant of the discriminant of the homogeneous equation of the general equation.
