Describing a Conic Section?

📌 CONCEPT: A conic section is a curve obtained by intersecting a cone with a plane, and the shape of the section depends on the angle at which the plane cuts the cone.

📐 RULE / FORMULA: The shape of the section is determined by the angle between the plane and the axis of the cone, which is given by the parameter m in the equation of the plane y = mx.

💡 WORKED EXAMPLE: Suppose a plane intersects the cone at an angle such that m = 1. The equation of the plane becomes y = x. As the plane approaches the y-axis, the shape of the section becomes a parabola. To show this, we can rewrite the equation of the plane as x^2 - y^2 = 0, which is the standard form of a parabola. This demonstrates that as the plane approaches the y-axis, the section of the cone becomes a parabola.

⚠️ COMMON MISTAKE: Students often assume that the shape of the section is always an ellipse or a circle, regardless of the angle at which the plane cuts the cone. However, the shape of the section can be a parabola, a hyperbola, or an ellipse, depending on the angle between the plane and the axis of the cone.