Cross Section | Meaning| Definition |Types of Cross section| Examples - Free Printable

The image you've uploaded illustrates conic sections, which are curves obtained by intersecting a double-napped cone with a plane. These conic sections include:

1. Circle
2. Ellipse
3. Parabola
4. Hyperbola

Let’s go through each one and explain how they are formed and their properties.

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🔷 Conic Sections – Overview

A cone (specifically a double-napped right circular cone) is formed by rotating a line (generator) around an axis. When a plane intersects this cone, the shape of the intersection depends on the angle of the plane relative to the cone's axis.

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1. Circle

- Formation: When the intersecting plane is perpendicular to the axis of the cone.
- Shape: A perfect round curve.
- Special Case: A circle is a special type of ellipse where the two foci coincide at the center.
- Equation (Standard Form):
$$
x^2 + y^2 = r^2
$$

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2. Ellipse

- Formation: When the plane intersects the cone at an angle less than the side of the cone but not perpendicular.
- Shape: An oval-shaped curve.
- Key Property: The sum of distances from any point on the ellipse to two fixed points (foci) is constant.
- Equation (Standard Form):
$$
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
$$
(where $ a > b $ for horizontal major axis)

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3. Parabola

- Formation: When the plane is parallel to the side (generator) of the cone.
- Shape: An open U-shaped curve.
- Key Property: Every point on the parabola is equidistant from a fixed point (focus) and a fixed line (directrix).
- Equation (Standard Form):
$$
y^2 = 4ax
$$
(opens right; other orientations possible)

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4. Hyperbola

- Formation: When the plane cuts both nappes of the cone at an angle steeper than the generator.
- Shape: Two separate open curves (branches).
- Key Property: The absolute difference of distances from any point on the hyperbola to two fixed points (foci) is constant.
- Equation (Standard Form):
$$
\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1
$$
(horizontal transverse axis)

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🔄 Summary Table

| Conic Section | Plane Angle Relative to Axis | Shape | Equation Example |
|---------------|-------------------------------|-------|------------------|
| Circle | Perpendicular | Round | $ x^2 + y^2 = r^2 $ |
| Ellipse | Oblique (not parallel to side)| Oval | $ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $ |
| Parabola | Parallel to generator | Open curve | $ y^2 = 4ax $ |
| Hyperbola | Steeper than generator | Two branches | $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $ |

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✔ Conclusion

This diagram visually demonstrates how all four conic sections arise from slicing a cone in different ways:
- Circle → Horizontal cut
- Ellipse → Slanted cut (not too steep)
- Parabola → Cut parallel to the side
- Hyperbola → Cut through both nappes

These curves have wide applications in physics (e.g., planetary orbits), engineering, optics, and more.

Let me know if you'd like derivations or real-world examples!