Conic sections are curves that result from the intersection of a plane with a double-napped cone (two cones stacked tip-to-tip). Depending on the angle and position of the intersecting plane, different types of curves—known as conic sections—are formed. These include circles, ellipses, parabolas, and hyperbolas. Each conic section has unique geometric properties and equations that describe it.

1. Circle

• Definition: A special case of an ellipse where all points on the curve are equidistant from a central point.
• Equation: $x^2 + y^2 = r^2$ (centered at the origin with radius $r$)
• Formation: When the plane is perpendicular to the cone’s axis and does not pass through the apex.

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2. Ellipse

• Definition: A closed curve where the sum of the distances from any point on the curve to two fixed points (called foci) is constant.
• Equation: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (for ellipses centered at the origin)
• Formation: When the plane cuts through the cone at an angle but doesn’t intersect both nappes.

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3. Parabola

• Definition: A curve where every point is equidistant from a fixed point (the focus) and a fixed line (the directrix).
• Equation: $y^2 = 4ax$ or $y = ax^2 + bx + c$ (depending on orientation)
• Formation: When the plane is parallel to a slant edge of the cone.

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4. Hyperbola

• Definition: A curve consisting of two disconnected branches. The difference in distances from any point on the hyperbola to two fixed points (foci) is constant.
• Equation: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
• Formation: When the plane cuts through both nappes of the cone.

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Degenerate Conics

• Point: If the plane passes only through the apex of the cone.
• Line or Pair of Lines: If the plane coincides with or passes through the edge of the cone.

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Summary of Key Properties

1. Circle: All points are equidistant from the center.
2. Ellipse: Sum of distances from two foci is constant.
3. Parabola: Focus and directrix define the shape.
4. Hyperbola: Difference in distances to two foci is constant.

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Would you like me to provide more details or specific examples?

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Here are 5 related questions:

1. What is the eccentricity of a conic section, and how does it determine the type of curve?
2. How does the general quadratic equation represent all conic sections?
3. What are the differences between an ellipse and a hyperbola?
4. How can parabolas be applied in real-world scenarios, such as satellite dishes or headlights?
5. What happens when the plane intersects the cone at different angles relative to the cone's axis?

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Tip: To determine the type of conic section from a quadratic equation, look at the discriminant $B^2 - 4AC$ in the general form $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$.