Q.
When the plane cuts the nappe (other than the vertex) of the cone, we have the following situations
Column I
Column II
A
When $\beta=90^{\circ}$
1
Section is a parabol
B
When $\alpha<\beta<90^{\circ}$
2
Section is a circle.
C
When $\beta=\alpha$
3
Section is an ellipse.
D
When $0 \leq \beta<\alpha$
4
Plane cuts through both the nappes and the curve of intersection is a hyperbola.
Here, $\beta$ is the angle made by the intersecting plane with the vertical axis of cone and $\alpha$ is the angle between the generator and vertical axis of cone.
Then, match the terms of Column I with terms of Column II and choose the correct option from the codes given below.
| Column I | Column II | ||
|---|---|---|---|
| A | When $\beta=90^{\circ}$ | 1 | Section is a parabol |
| B | When $\alpha<\beta<90^{\circ}$ | 2 | Section is a circle. |
| C | When $\beta=\alpha$ | 3 | Section is an ellipse. |
| D | When $0 \leq \beta<\alpha$ | 4 | Plane cuts through both the nappes and the curve of intersection is a hyperbola. |
Conic Sections
Solution:
A. When $\beta=90^{\circ}$; the section is a circle.
B. When $\alpha<\beta<90^{\circ}$; the section is an ellipse.
C. When $\beta=\alpha$; the section is a parabola.
D. When $0 \leq \beta<\alpha$; the plane cuts through both the nappes and the curves of intersection is hyperbola.
(in each of $A, B$ and $C$, the plane cuts entirely across one nappe of the cone).
