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The Conics

By: Diana Brown

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Day Seven:

Parabolas

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Parabolas:

The focus lies on the line of symmetry

The directrix is perpendicular to the line of symmetry.

The vertex lies halfway between the focus and the directrix.

Diagram:

Standard equations of a parabola (vertex at origin)

Equation	Focus	Directrix	Axis of Symmetry
x² = 4py	(0, p)	Y = - p	X = 0
y² = 4px	(p, 0)	X = - p	Y = 0

More equations of the parabola are as follows:

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Sample Problems

1. Identify the focus and directrix of the parabola given by x = - ¼ y². Graph the parabola.

Solution:

First solve the equation for y²:

y²= -4x

The parabola opens left. 

To find p:

4p = -4 and p = -1

Therefore the focus is (-1, 0) and the directrix is x = -(-1) = 1.

2. Find the equation of the parabola with vertex at (0, 0) and directrix y = 2.

Solution:

                Since the directrix is a horizontal line and is above the vertex, the parabola opens down.  p = 2 (distance from directrix to vertex), so 4p = 8.  Thus the equation is y = -(1/8)x²

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Using parabolas in real life:

Parabolic reflectors have cross sections that are parabolas. A special property of any parabolic reflector is that all incoming rays parallel to the axis of symmetry that this the reflector are directed to the focus. Similarly, rays emitted from the focus that hit the reflector are directed in rays parallel to the axis of symmetry. These properties are the reason satellite dishes and flashlights are parabolic.

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Go to Day Eight (Hyperbola Introduction)

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