Conic Sections and Ellipses

• Conic sections include circles, ellipses, parabolas, and hyperbolas, and were discovered by Menaechmus, with contributions from Aristaeus, Euclid, and Archimedes.
• All conic sections have either one or two foci.
• A focus is a point where waves bouncing off a curve converge, seen in satellite dishes.
• Ellipses and hyperbolas have two foci, while circles and parabolas have only one.
• Ellipses have a plus sign between the two fractions in their formula, while hyperbolas have a minus sign.

Ellipses

• An ellipse is a set of points where the sum of the distances from two focal points to any point on the ellipse is constant (FP + GP = constant).
• Ellipses have two vertices, which are the endpoints of the major axis (longest diameter), and two covertices, which are the endpoints of the minor axis (perpendicular to the major axis).
• The equation of a horizontal ellipse is where (h, k) is the center, a is the major radius, and b is the minor radius.
• The equation of a vertical ellipse is where (h, k) is the center, a is the major radius, and b is the minor radius.
• Ellipsoid shapes in whispering galleries focus whispers from one focus to the other.

Hyperbolas

• A hyperbola is a set of points where the absolute value of the difference between the distances from two focal points to any point on the hyperbola is constant (|FP - GP| = constant).
• Hyperbolas consist of two parabola-like shapes with two vertices.
• The equation of a horizontal hyperbola is , where (h, k) is the center, a is the distance from the center to a vertex, and b is the positive asymptote slope multiplied by a.
• The equation of a vertical hyperbola is where (h, k) is the center, a is the distance from the center to a vertex, and b is the reciprocal of the positive asymptote slope divided by a.

Foci of Ellipses

• The foci of an ellipse are where waves bouncing within the ellipse converge.
• The focal distance, c, relates to the major radius, a, and the minor radius, b, by the equation .
• Foci are equidistant from the center and lie along the major axis.
• The foci of a horizontal ellipse with center (2, -1) and focal distance 2 are (0, -1) and (4, -1).

How to find the foci of an ellipse

• Find the midpoint of the vertices, or the intersection of the major and minor axes to get the center point, C.
• Calculate the major radius, a, as the distance between point C and a vertex.
• Calculate the minor radius, b, as the distance between point C and a covertex.
• Use to solve for c.
• Locate the foci along the major axis, c units from the center.
• For an ellipse with vertices/covertices at (0, 2), (10, 2), (5, 5), and (5, -1), the center is (5, 2), a = 5, b = 3, c = 4, and the foci are (1, 2) and (9, 2).

Foci of Hyperbolas

• Foci are points where line segments to any point on the hyperbola have a constant length difference.
• The center point is the midpoint between the two vertices.
• Foci are equidistant from the center and lie directly past the vertices.
• For a horizontal hyperbola with a center at (2, -1) and a focal distance of 3, the foci are (-1, -1) and (5, -1).
• The distance from the center to either focus is represented by c.

How to find the foci of a hyperbola

• Determine if the hyperbola is horizontal or vertical based on the vertices.
• Find the midpoint of the vertices to determine the center point, C.
• Determine the focal distance, c, which is half the distance between the foci.
• Move c units from point C along the axis to locate the foci.
• For a hyperbola with vertices at (2, 3) and (2, 7) and foci 10 units apart, it's a vertical hyperbola with center (2, 5), c = 5, and foci at (2, 0) and (2, 10).