In the figure

Statement-I: When \(\alpha > \beta \geq 0\), the section is hyperbola
Statement-II: When \(\beta > 90^\circ\), the section is ellipse
Which of the following is correct?

Let \(\alpha\) be the semi-vertical angle of the cone and \(\beta\) be the angle between the intersecting plane and the axis of the cone.

The conic sections formed by the intersection of a plane and a double cone are determined by the relationship between \(\alpha\) and \(\beta\):
1. If \(0 \le \beta < \alpha\), the plane intersects both nappes of the cone, forming a hyperbola.
2. If \(\beta = \alpha\), the plane is parallel to a generator of the cone, forming a parabola.
3. If \(\alpha < \beta < 90^\circ\), the plane intersects only one nappe, forming an ellipse.
4. If \(\beta = 90^\circ\), the plane is perpendicular to the axis, forming a circle.

From the above conditions, when \(\alpha > \beta \ge 0\), the section is a hyperbola. Hence, Statement-I is true.

The angle \(\beta\) is defined as the angle between the plane and the axis, which is taken in the range \(0^\circ \le \beta \le 90^\circ\). Statement-II claims that for \(\beta > 90^\circ\) the section is an ellipse, which is incorrect. Thus, Statement-II is false.

Therefore, Statement I is true and Statement II is false.

Answer: Statement I is true, Statement II is false