Ellipses, Parabolas, Hyperbolas: The Puzzle of The Family of Curves

Introduction:

The formulas for expressing ellipses, parabolas, and hyperbolas are based on their geometric properties:

• Ellipses are expressed by the formula:

\( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 2 \),

where\(a\)  and\(b\)  are the distances from the center to the ellipse’s vertices along the major and minor axes, respectively. Ellipses are found in planetary orbits due to gravitational forces, and their formula is used in calculating orbits and designing satellite paths.

• Parabolas have the formula:

\( y = ax^3 + bx + c \),

in their simplest form, where\(a, b\),  and\(c  are constants that determine the shape and position of the parabola. This shape is observed in the path of projectiles under gravity, and the formula is used in designing structures like bridges and satellite dishes, where reflecting properties of parabolas are essential.

• Hyperbolas are represented by:

\( \frac{x^2}{a^2} – \frac{y^2}{b^2} = 2 \) or \( \frac{y^2}{b^2} – \frac{x^2}{a^2} = 2 \),

depending on their orientation. They are found in the paths of objects in hyperbolic motion and are used in navigation systems and to describe the behavior of certain types of lenses.

Each of these conic sections is derived from slicing a cone at different angles and positions, and their mathematical properties are used in various fields of science, engineering, and technology.

Here is an example:

The formula for an ellipse is based on its geometric properties and is given by:

\( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \),

where\(x\)  and\(y\)  are coordinates on the ellipse,\(a\)  is the semi-major axis, and\(b\)  is the semi-minor axis.

Explanation:

1. The ellipse is a conic section, a curve obtained by slicing a cone with a plane.

2. The equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)  represents a set of all points\((x, y) \)  such that the sum of the distances from these points to two fixed points (foci) is constant.

3. The semi-major axis,\(a\),  is half the longest diameter of the ellipse, and the semi-minor axis,\(b\),  is half the shortest diameter.

4. In this equation, the center of the ellipse is at the origin\((0, 0)\)  of the coordinate system. The lengths\(2a\) and \(2b\)  are the lengths of the ellipse along the \(x-axis\) and \(y-axis\), respectively.

5. The ellipse is symmetric with respect to both the \(x-axis\) and the \(y-axis\). This symmetry is reflected in the equation’s structure.