PARABOLA
A parabola is a curve where any point is at an
equal distance from a fixed point (the focus ),
and a fixed straight line (the directrix )
FUNDAMENTAL ELEMENTS OF A PARABOLA
The following are the fundamental elements
of a parabola:
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Vertex
Focus
Focal length
Latus rectum
Directrix
Axis
VERTEX
▪ The vertex of the parabola is its extreme point. If the
parabola opens upwards, the vertex represents the
lowest point in the parabola. If the parabola opens
downwards, the vertex represents the highest point.
In either case, the vertex is a point that changes the
direction of the parabola. Frequently, the vertex is
represented with the letter V.
FOCUS
▪ The focus is a fixed point used to define the
parabola. This point is not located on the parabola,
but inside. The focus is denoted by F.
FOCAL LENGTH
▪ The focal length is the length between the vertex and
the focus.
LATUS RECTUM
▪ The latus rectum is a line perpendicular to the line
joining the vertex and the focus and is four times the
length of the focal length.
DIRECTRIX
▪ The directrix is a straight line in front of the parabola.
We use d to represent the directrix. The distance
between the directrix and the vertex is the same as the
distance between the focus and the vertex.
AXIS
▪ The axis of the parabola is a line perpendicular to the
directrix. The axis represents the line of symmetry of
the parabola.
HYPERBOLA
A hyperbola is the set of all points, which have distances
from two fixed points, called foci, which have a difference
that is equal to a constant.
We can also define hyperbolas as the conic sections that
are formed by the intersection of two cones with an
inclined plane that intersects the base of the cones.
Hyperbolas consist of two separate curves, called
branches.
FUNDAMENTAL ELEMENTS OF A HYPERBOLA
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Foci
Transverse axis
Conjugate axis
Semi-major axis
Semi-minor axis
Center
Vertices
Focal length
Axes of symmetry
Asymptotes
FOCI
▪ The foci are the fixed points used to define the hyperbola.
The foci are often defined by F1 and F2 or also by F and F’.
TRANSVERSE AXIS
▪ The transverse axis, also known as the real axis, is the
segment that goes through the two foci. The transverse
axis can be determined using the equation of the
hyperbola.
CONJUGATE AXIS
▪ The conjugate axis, also known as the imaginary axis, is
the perpendicular bisector of the transverse axis. The
conjugate axis divides the transverse axis into two equal
parts.
SEMI-MAJOR AXIS
➢ The semi-major axis is the segment that
extends from the center to a vertex of the
hyperbola. Its length is denoted by a.
SEMI-MINOR AXIS
▪ The semi-minor axis is the segment
perpendicular to the semi-major axis. Its
length is denoted by b.
CENTER
▪ The center has two lines of symmetry. The
center is the point of intersection of the two
lines of symmetry. If the hyperbola is centered
at the origin, the center is (0,0) and if it is
centered at another point, the center is (h,k)
VERTICES
• The vertices are the points of intersection of the hyperbola
with the transverse axis. The vertices are the endpoints of
each branch of the hyperbola. Usually, we use V1 and V2 or V
and V’ to represent the vertices.
FOCAL LENGTH
• The focal length is the length of the segment that extends
from one focus (F1) to the other (F2). Its length is equal to 2c.
AXES OF SYMMETRY
• The lines of symmetry are the axes that coincide with the
transverse axis and the conjugate axis. The two branches of
the hyperbola are symmetrical. Hyperbolas have two lines
of symmetry, the horizontal axis, and the vertical axis. The
point of intersection of these axes is the center.
ASYMPTOTES
• The asymptotes are the lines that are very close to the
branches of the hyperbola but never touch it. The
asymptotes intersect at the center of the hyperbola.