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

Step 1: Understanding the Concept:
The problem relates to the geometric definition of conic sections obtained by intersecting a right circular double cone with a plane. The type of conic section depends on the relationship between the semi-vertical angle of the cone ($\alpha$) and the angle the intersecting plane makes with the vertical axis of the cone ($\beta$).

Step 2: Key Formula or Approach:
Recall the standard classification of conic sections based on the angles $\alpha$ and $\beta$: - If $\beta = 90^\circ$, the intersecting plane is perpendicular to the axis, and the section is a circle. - If $\alpha<\beta<90^\circ$, the plane cuts entirely across one nappe of the cone, and the section is an ellipse. - If $\beta = \alpha$, the plane is parallel to a generator of the cone, and the section is a parabola. - If $0 \le \beta<\alpha$, the plane intersects both nappes of the double cone, resulting in a two-part curve called a hyperbola.

Step 3: Detailed Explanation:
Let's evaluate each statement based on the rules above. Statement I: "When $\alpha>\beta \ge 0$, the section is hyperbola" This condition can be rewritten as $0 \le \beta<\alpha$. According to our classification, when the plane's angle with the axis is less than the semi-vertical angle, it cuts both halves of the cone, forming a hyperbola. Therefore, Statement I is true. Statement II: "When $\beta = 90^\circ$, the section is ellipse" When $\beta = 90^\circ$, the plane is exactly horizontal (perpendicular to the vertical axis). The resulting intersection is perfectly symmetric around the axis, forming a circle. While a circle is technically a special, degenerate case of an ellipse (where eccentricity $e=0$), in the context of standard conic section classification problems, "ellipse" refers to the general case where $\alpha<\beta<90^\circ$ and the cross-section is elongated. A circle is treated as a distinct category. Thus, stating it is an ellipse when it is specifically a circle makes the statement false by convention in such multiple-choice questions.

Step 4: Final Answer:
Statement I is true and Statement II is false, which corresponds to option (1).