Q.A double right circular cone with semi-vertical angle $\alpha$ is cut by a plane passing through the vertex $V$. State the degenerate conic section obtained in each case:

Every conic section -- circle, ellipse, parabola or hyperbola -- is obtained by intersecting a plane with a double right circular cone, and which curve results depends entirely on the angle the cutting plane makes with the cone's axis compared with the cone's own semi-vertical angle: a plane perpendicular to the axis gives a circle, one tilted between the axis-angle and the perpendicular gives an ellipse, one exactly parallel to a generator gives a parabola, and one steep enough to cut both nappes gives a two-branched hyperbola. If the same cutting plane is instead made to pass through the cone's vertex, each of these four ordinary conics collapses to a degenerate limiting case: the circle/ellipse family degenerates to a single point, the parabola family to a single straight line, and the hyperbola family to a pair of intersecting straight lines through the vertex. This geometric origin is what unifies four seemingly different curves into one family, and its eccentricity value ($e=0$ circle, $0<e<1$ ellipse, $e=1$ parabola, $e>1$ hyperbola) gives the same classification purely algebraically.

Conic sections form their own NCERT Class 11 Mathematics chapter, and this geometric-origin explanation is a common CBSE board and JEE Main conceptual question. Students searching "conic sections definition class 11 maths" or "how are conic sections formed NCERT" will find this concept answers exactly that.