**PARABOLAS**

ELLIPSES

CIRCLES

**Real Life Applications of Conic Sections**

Definition

An ellipse is a regular oval shape that has two focal points in which the sum of the length of both focal points to any given point on the ellipse is always the same. Also, any line connected to the outside of the ellipse from one focal point will immediately be reflected back through the other focal point. An ellipse is created when a cone is cut by an oblique plane that does not intersect the base.

Definition

A circle is formed by cutting a circular cone with a plane perpendicular to the symmetry axis of the cone. It's a round plane figure whose boundary (the circumference) consists of points equidistant from a fixed point (the center).

Application

Pizzas are made in the shape of circles. The radius of the pizza is created by slicing from the center to any point on the circumference of the pizza. No matter which way you slice, the radius will always be the same length.

Equation:

Measurements

Equations:

Application

The orbit of Earth around the Sun is in the shape of an ellipse. In this case, the Sun is recognized as one of the two focal points within the ellipse.

Measurements

a = 149,700,000 km

c = 2,400,000 km

To find b:

c^2 = a^2 - b^2

b = 149,680,760.3 km

Length major axis: 299,400,000 km

Length minor axis: 299,361,520.6 km

Center: (0,0)

Vertices: (149.7 mil.,0),(-149.7 mil.,0)

Foci: (2.4 mil,0),(-2.4 mil.,0)

Equation:

Radius = 8 in.

Center: (0,0)

Equation:

Horizontal

Vertical

Definition

A hyperbola is a symmetrical open curve formed when a plane slices the top and bottom section of a cone.

Equations:

Horizontal Transverse Axis

Vertical Transverse Axis

Application

Lightbulbs inside lampshades project hyperbolic curves on the wall behind them. The center of the hyperbola, in this case, is the light bulb.

Measurements

Height (Length of transverse

axis OR 2a) = 10 in.

Diameter (Length of conjugate axis OR 2b) = 8 in.

Therefore:

a = 5 in.

b = 4 in.

To find c:

c^2 = a^2+ b^2

c = 6.4 in.

Center: (0,0)

Vertices: (0,5),(0,-5)

F1: (0,6.4)

F2:(0,-6.4)

Asymptotes: y = +5/4x, y = -5/4x

Equation:

Definition

Equations:

Up or Down

Left or Right

Application

Measurements

A parabola is formed by intersecting the plane through the cone and the top of the cone. A parabola is the set of points that are equally distant from a focus point and the directrix, a fixed line.

Equation:

The Golden Gate Bridge is a suspension bridge that utilizes a parabolic structure. At first glance, the curve may be described as a catenary. But since the curve of a suspension bridge is not created by gravity alone (the forces of compression and tension are acting on it) it can't be considered a catenary, but rather a parabola.

Height: 500 ft

Width: 4,200 ft

Vertex: (0,0)

To find p:

Plug coordinates

into x^2=4py

p = 2,205, 4p= 8,820

Focus point: (0,2205)

Directrix: y = -2,205