Calculus
Lesson
58
Conics
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FOUR types of conics, all defined by
Circle:
Center: 
, Radius: 
The definition of a circle is the locus of all points that are a
distance 
away from
the center.
Ellipse: 
horizontal
vertical
The definition of an ellipse is the locus of all points that are a combined
distance 
away from
the two foci.
is the major axis (big one) and
is the
minor axis (small one)
Center: 
Vertices are located 
and
from the center.
Foci are located 
away
from the center, 
Eccentricity: 
(note 
is a circle,
is a line)
Example: Find everything about the ellipse defined by:
Complete the square twice to get 
Center: 
Vertices: 
Foci: 
Area of an ellipse: 
(4
for each quadrant separately).
NO WAY to solve for circumference! 
Not integrable!
Parabola:
vertical
horizontal
Definition of a parabola: the locus of all points that are equidistant from a line and a
point 
away from
the line.
Vertex: 
is the distance from
the vertex to the directorix.
is also the distance
from the vertex to the focus.
The "latus rectum" is the line parallel to the directorix that passes through
the focus.
Any light ray that comes in perpendicular to the directorix MUST pass
through the focus, which is why we have parabolic mirror in telescopes.
Hyperbola: 
horizontal
vertical
The definition of a hyperbola is the locus of all points whose difference in distances
between the point and the foci is a constant, 
.
Center: 
The foci are 
away from
the center, where 
is the distance from
the center to the vertex along the hyperbolic axis.
is the distance between
vertices.
is the width of the
imaginary box, the diagonals of which are the
hyperbolic asymptotes, along the conjugate axis.
