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A hyperbola is a type of conic section formed by the intersection of a plane with two cones (placed point to point) that results in two disconnected curves. This is distinct because unlike ellipses or circles which form closed curves, hyperbolas are open-ended.
In mathematical terms, the standard form of a hyperbola can be either:

• Horizontal: \ \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)
• Vertical: \ \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \)

The key characteristic of a hyperbola is the terms having opposite signs, as seen in equations \ \(-x^2 + y^2 = 25\) or the rearranged form equivalent to \ \(\frac{-x^2}{25} + \frac{y^2}{25} = 1\).
This specific arrangement indicates a vertical hyperbola where the curve "opens" upward and downward. The positive term's variable dictates the direction of the opening.

A parabola is another fundamental conic section. It's the result of a plane intersecting a cone at a slant parallel to one of its sides. Parabolas open either like a "U" or an inverted "U", and are exemplified by their vertex as the point where the curve changes direction.
The standard equations for parabolas include:

• Vertical: \ \( y = ax^2 + bx + c \)
• Horizontal: \ \( x = ay^2 + by + c \)

When dealing with parabolas, it's important to recognize their distinctive "U" shape and that they do not close back on themselves like circles or ellipses. The variable squared indicates the direction of the parabola: if \ \(x\) is squared, the parabola opens vertically, and if \ \(y\) is squared, it opens horizontally.

An ellipse is a conic section that resembles a flattened circle or an elongated shape. It is formed by intersecting a plane with a cone at an angle that is less steep than the cone's side but not parallel to the base.
Ellipses are characterized by:

• Horizontal Major Axis: \ \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) where \ \(a > b\)
• Vertical Major Axis: \ \( \frac{y^2}{b^2} + \frac{x^2}{a^2} = 1 \) where \ \(b > a\)

Ellipses have two foci (plural of focus), which are points inside the ellipse such that the sum of the distances from these foci to any point on the ellipse remains constant. This unique property helps in identifying and distinguishing ellipses from other conic sections.

A circle is often considered a special case of an ellipse where both major and minor axes (the distances through the center) are equal. The intersection of a horizontal plane with a cone creates a perfect circle.
Mathematically, the simplest form of a circle's equation is:

• \ \( x^2 + y^2 = r^2 \)

Here, \ \(r\) represents the radius of the circle, which is constant and the same in all directions from the center point.
Circles are unique among conic sections as they are symmetric around their center, with no variation in width or height. Recognizing a circle involves recognizing this symmetry and the lack of both broader horizontal or vertical axes which would indicate elliptical shape.