# Conic Sections: A Guide to Slicing a Cone
## An Overview of Circles, Ellipses, Parabolas, and Hyperbolas

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## 🎯 Introduction: The Carrot and the Knife

Have you ever wondered about the different shapes you can get by slicing a carrot (which is shaped like a cone) with a knife? Depending on the angle of your cut, you can create a circle, an oval (ellipse), and other interesting curves. These shapes are known as **conic sections**. They are fundamental curves in geometry and appear everywhere in science and nature. 🥕

This guide explains how these four key shapes are formed by slicing a cone.

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## 🔑 What is a Cone?

In geometry, when we talk about a cone for conic sections, we're usually referring to a **double-napped right circular cone**.

Imagine two fixed lines, AB and CD, that intersect at a point **O**. If you rotate the line AB around the fixed line CD, the surface you create is this double cone.

![Diagram of a double-napped right circular cone](https://seekoguide.com/images/conic/1.svg)

Here are the key parts:
* **Vertex (O):** The point where the two cones meet.
* **Axis (CD):** The fixed line that the other line rotates around.
* **Generator (AB):** The moving line that traces the surface of the cone.
* **Nappes:** The two separate cones are called nappes (an upper nappe and a lower nappe).

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## 🔪 Creating Shapes by Slicing the Cone

The four conic sections are created by slicing this double cone with a flat plane at different angles.

### 1. The Circle
When you slice the cone with a plane that is **perfectly horizontal** (perpendicular to the axis) and parallel to the circular base, the resulting shape is a **circle**.

![Formation of a circle by slicing a cone](https://seekoguide.com/images/conic/circle.svg)

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### 2. The Ellipse
If you tilt the plane slightly, so the slice is **not parallel to the base** but still cuts completely through only one nappe, the resulting shape is an **ellipse**. An ellipse is essentially a stretched or squashed circle.

![Formation of an ellipse by slicing a cone](https://seekoguide.com/images/conic/ellipse.svg)

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### 3. The Parabola
When you slice the cone with a plane that is **parallel to one of the generator lines** (the slanted edge of the cone), the curve you get is a **parabola**. This curve does not close back on itself.

![Formation of a parabola by slicing a cone](https://seekoguide.com/images/conic/parabola.svg)

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### 4. The Hyperbola
If you slice the cone with a plane that is steep enough (parallel to the axis or at an even steeper angle) to cut through **both the upper and lower nappes**, you get a **hyperbola**. A hyperbola consists of two separate, mirror-image curves.

![Formation of a hyperbola by slicing a cone](https://seekoguide.com/images/conic/hyperbola.svg)

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## ✍️ Exercises

### Question 1: Parts of a Cone
Name the different parts of the cone indicated by the letters a, b, c, d, and e in the figure below.

![Cone with labeled parts for exercise 1](https://seekoguide.com/images/conic/one.svg)

**Answer:**
* **a:** Axis
* **b:** Upper Nappe
* **c:** Lower Nappe
* **d:** Generator
* **e:** Vertex

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### Question 2: Identify the Conic Section
Name the conic section shown in each of the following figures.

1.  ![Diagram for exercise 2a](https://seekoguide.com/images/conic/a.svg)
    **Answer:** Ellipse

2.  ![Diagram for exercise 2b](https://seekoguide.com/images/conic/b.svg)
    **Answer:** Parabola

3.  ![Diagram for exercise 2c](https://seekoguide.com/images/conic/c.svg)
    **Answer:** Circle

4.  ![Diagram for exercise 2d](https://seekoguide.com/images/conic/d.svg)
    **Answer:** Hyperbola