How Conic Sections are Formed: Intersecting a Cone with a Plane

📚 Introduction to Conic Sections

Conic sections are curves formed when a plane intersects a double cone. The type of curve formed depends on the angle and location of the intersection.

🎯 Objectives

• 🧭 Understand the definition of a conic section.
• ✂️ Identify the four main types of conic sections.
• 📐 Explain how different plane intersections create different conics.

🛠 Materials

• 📃 Worksheet with diagrams of cones and planes
• ✏️ Pencils
• 📏 Rulers
• 🧱 Modeling clay and a knife (optional, for a hands-on demonstration)

🔥 Warm-up (5 mins)

Ask students what shapes they know and where they see them in everyday life. Briefly review the concepts of planes and cones.

👨‍🏫 Main Instruction

1. ⭕ Circle

   • 📐 The plane is perpendicular to the axis of the cone.
   • 🔍 All points on the circle are equidistant from the center.
   • 📝 Equation: $(x-h)^2 + (y-k)^2 = r^2$

2. 💫 Ellipse

   • 📐 The plane is tilted relative to the axis of the cone, but not parallel to the side.
   • 🔍 The sum of the distances from any point on the ellipse to two foci is constant.
   • 📝 Equation: $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

3. 🎢 Parabola

   • 📐 The plane is parallel to the side of the cone.
   • 🔍 All points on the parabola are equidistant from a focus and a directrix.
   • 📝 Equation: $y = ax^2 + bx + c$ or $x = ay^2 + by + c$

4. hyperboles

   • 📐 The plane is parallel to the axis of the cone, intersecting both halves of the double cone.
   • 🔍 The absolute difference of the distances from any point on the hyperbola to two foci is constant.
   • 📝 Equation: $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$ or $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$

✅ Assessment

Complete the following quiz to assess your knowledge of conic sections.

❓ Practice Quiz

1. ❓ Which conic section is formed when the plane is perpendicular to the axis of the cone?
2. ❓ Which conic section is formed when the plane is parallel to the side of the cone?
3. ❓ Which conic section is formed when the plane intersects both halves of the double cone?
4. ❓ What is the key property of a circle?
5. ❓ What is the key property of a parabola?
6. ❓ What is the key property of an ellipse?
7. ❓ What is the key property of a hyperbola?