Circumference, Ellipse, Parabola, and Hyperbola

Posted on Feb 14, 2025 in Mathematics

Properties of the Circumference

The circumference is generated by the intersection of a cylinder or a straight circular cone with a plane. Geometrically, the circumference has several elements, such as:

• Center: A fixed point equidistant to all points of the circumference.
• Radius: The distance of the segment that joins the center of the circumference to any of its points.
• Chord: The segment that joins any two points of the circumference.
• Diameter: The chord that passes through the center and has the maximum length.
• Secant: The line that cuts any two points of the circumference.
• Tangent: The line that touches the circumference at one point.
• Point of tangency: The point where the tangent line touches the circumference.
• Arc: The curvilinear segment formed by different points of the circumference.

These elements are shown in the following figure:

Example: Calculating a Painted Area

You were asked to paint a section of a mural, represented as the shadowed area in the following figure. Calculate the area that must be painted.

Steps

1. Procedure: The area to paint can be obtained by using the following formula:
2. Calculate the largest area.
3. Calculate the smallest area.

Answer

Therefore, the area you should paint is:

Properties of the Ellipse

The ellipse is obtained by the intersection of a straight circular cone and a plane not parallel to the base of the cone.

The elements of the ellipse are:

• Major axis: The segment that goes from one vertex to another and measures 2a.
• Minor axis: The segment perpendicular to the major axis.
• Focal axis: The segment between the two foci.
• Foci: Two fixed points of the ellipse.
• Vertices: Points where the ellipse axis cuts.
• Center: A point equidistant from the foci and the intersection between the major and minor axis.

Example: Football Trajectory

A football player kicks a ball. The play is video recorded by a fan that is a Physics student. When arriving home, he studies the ball’s path and deduces that it describes an ellipse. Determine the coordinates of the foci of the ellipse by referring to the following figure:

Steps

1. Procedure:
   • If the major axis measures 22 meters, then:

     V´V = 2a = 22

     a = 11

   • If the minor axis measures 10 meters, then:

     F´F = 2b = 10

     b = 5

   • Calculate variable c:

     c² = a² – b²

     

Answer

The center of the ellipse is at (0, 11), and the coordinates of the foci are: F(0, 11 ± 9.8) => F’(0, 1.2) and F(0, 20.8)

Properties of the Parabola

The parabola is a conic section that corresponds to the slice of a straight circular cone parallel to any element of the cone.

The elements of the parabola are:

• Focus
• Directrix
• Parabola axis
• Latus rectum

Example: Bridge Arch Equation

One of the most common applications of the parabola is for building bridges. Determine the equation of the arch according to the data stated by the following figure:

Steps

1. Procedure: Using the equation of the parabola that opens upward and taking as reference that the uppermost point of the post at the right has coordinates (250, 150), we have that:

Answer

The equation of the arch is:

Properties of the Hyperbola

The elements of the hyperbola are:

• Major axis (Transverse axis): The segment that goes from one vertex to the other and measures 2a.
• Minor axis (Conjugate axis): The segment perpendicular to the major axis.
• Focal axis: The segment between the two foci.
• Foci: Two fixed points of the hyperbola.
• Vertices: Points that cut the axis of the hyperbola.
• Center (Center of symmetry): The midpoint between the foci. It is also the intersection between the major and minor axis.
• Latus rectum: The chord perpendicular to the focal axis.

Example: Hyperbolic Towers

In an energy production plant, there are two towers with a hyperbolic form, as shown:

This hyperbola is generated in a third dimension. The narrowest part of the tower has a distance of 50 meters and an eccentricity of . Find the equation of the hyperbola.

Steps

1. Procedure:
   • The closest points in a hyperbola are its vertices, then:
   • From the definition of eccentricity, we have:
   • With values of a and c, we calculate the value for b to obtain the equation.

Answer

Therefore, the equation is: