Question

In the figure
Statement I : When $\alpha>\beta \ge 0$, the section is hyperbola
Statement II : When $\beta = 90^\circ$, the section is ellipse
Which of the following is correct?

• A. Statement I is true, Statement II is false
• B. Statement I is false, Statement II is true
• C. Both the Statements are true
• D. Both the Statements are false

Hint:

A good way to visualize this: as the plane tilts more vertically (smaller $\beta$), it eventually cuts through both the top and bottom cones. This multi-part intersection is the hallmark of a hyperbola.

Answer 1

The Correct Option is A

To solve this question, we must understand the properties of conic sections (ellipse, parabola, and hyperbola) formed by the intersection of a plane with a cone. The given statements describe scenarios with angles representing the inclination of the plane. Let's analyze each statement:

1. Statement I: When \(\alpha > \beta \ge 0\), the section is hyperbola.
   • In conic sections, a hyperbola is formed when the plane cuts through both nappes of the cone. This typically occurs when the angle \(\alpha\) (between the plane and the vertical axis of the cone) is greater than the angle \(\beta\) (the half-angle of the cone). Thus, when \(\alpha > \beta\), the plane cuts both nappes, and the section is indeed a hyperbola.
   • This statement is true.
2. Statement II: When \(\beta = 90^\circ\), the section is ellipse.
   • If the angle of the cone \(\beta = 90^\circ\), it implies that the cone opens completely flat like a plane, which is not a typical configuration for forming standard conic sections like ellipses. In a realistic scenario, such a configuration does not describe an ellipse but rather a degenerate form.
   • This statement is false.

Considering the analysis above, the correct option is: Statement I is true, Statement II is false.