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Transcript of CIRCLES
Real Life Applications of Conic Sections
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.
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).
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.
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.
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
Vertices: (149.7 mil.,0),(-149.7 mil.,0)
Foci: (2.4 mil,0),(-2.4 mil.,0)
Radius = 8 in.
A hyperbola is a symmetrical open curve formed when a plane slices the top and bottom section of a cone.
Horizontal Transverse Axis
Vertical Transverse Axis
Lightbulbs inside lampshades project hyperbolic curves on the wall behind them. The center of the hyperbola, in this case, is the light bulb.
Height (Length of transverse
axis OR 2a) = 10 in.
Diameter (Length of conjugate axis OR 2b) = 8 in.
a = 5 in.
b = 4 in.
To find c:
c^2 = a^2+ b^2
c = 6.4 in.
Asymptotes: y = +5/4x, y = -5/4x
Up or Down
Left or Right
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.
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
To find p:
p = 2,205, 4p= 8,820
Focus point: (0,2205)
Directrix: y = -2,205