The Conics
By: Diana Brown
Day Seven:
Parabolas
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:
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
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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Eight (Hyperbola Introduction)