Definition:Degenerate Conic

Theorem

A degenerate conic is a conic section whose slicing plane passes through the apex of the cone.

There are three possibilities:

A point-circle is the locus in the Cartesian plane of an equation of the form:

$(1): \quad \paren {x - a}^2 + \paren {y - b}^2 = 0$

where $a$ and $b$ are real constants.

There is only one point in the Cartesian plane which satisfies $(1)$, and that is the point $\tuple {a, b}$.

It can be considered to be a circle whose radius is equal to zero.

A degenerate ellipse is the conic section whose slicing plane passes through the apex of the cone.

Hence it consists of a single point.

A degenerate parabola is the conic section whose slicing plane passes through the apex of the cone and is thus tangent to the cone

Hence it consists of a single straight line.

Let $\phi < \theta$, that is: so as to make $K$ a hyperbola.

However, let $D$ pass through the apex of $C$.