Look at the family of parabolae y=ax^2 where a is the parameter.
Their graphs are nested into each other. All of them passes through the point (0,0). The smaller a is, the flatter the curve is.
A degenerated form would occur when a goes to 0 or to infinity.
If a approaches 0, the curve will become very flat. The degenerated curve will be the x-axis. It may be obtained by the intersection of a plane tangent to a double cone. It is not a solution to our problem!
Let's try a increasing to infinity. The branches of the parabolae will be closer an closer to the y-axis. The degenerated curve will be the positive part of the y-axis. It is a half line. It can't be obtained by the intersection of a plane and a double napped cone.