2. 2
hyperbola
when the plane (not necessarily
vertical) intersects both cones to form
two unbounded curves (each called a
branch of the hyperbola)
3. hyperbola
3
is a set of all points in a place such the difference
of the distances of each point of the set from the fixed
points is constant. The fixed point are called foci.
Since c>a, the number 𝒄𝟐
− 𝒂𝟐
is a positive; we let 𝒃𝟐
= 𝒄𝟐
−
𝒂𝟐
. Then we can write the equation of the Ellipse
𝑥2
𝑎2
−
𝑦2
𝑏2
= 1
5. Part of the hyperbola
5
Horizontal Hyperbola
Figure 1.3
𝑥2
𝑎2
−
𝑦2
𝑏2
= 1
𝑦2
𝑎2
−
𝑥2
𝑏2
= 1
Vertical Hyperbola
Figure 1.2
6. Definition of terms:
6
Focus- is called eccentricity of the hyperbola.
Vertices- the points of the hyperbola with the traverse axis.
Directrix (fixed line)- is a line such that the ratio of distance
of the points on the hyperbola from the focus to its
distance from the directrix is constant.
Latera Recta- the line segment passing through the foci
and perpendicular to the transverse axis.
Transverse axis- the line segment joining the vertices.
Conjugate axis- the segment which is perpendicular
bisector of the transverse axis with the length of 2b.
7. Definition of terms:
7
Focal length- is the line segment joining the foci with the
length of 2c.
Asymptote- the lines that pass to the center of the
hyperbola and are asymptotic to the curves.
Center- is the center where the two asymptotes intersect,
or it can be defined as the intersection of the traverse and
conjugate.
9. For simplicity, the following will be represented as;
9
𝑪= Center of the Hyperbola
𝑭𝟏, 𝑭𝟐 = Foci(Plural form of
focus)
𝑽𝟏, 𝑽𝟐= Vertices
𝑩𝟏, 𝑩𝟐 =Endpoint of the
conjugate axis
𝑷 𝒙, 𝒚 = any point along the
Hyperbola
𝑫𝟏, 𝑫𝟐= Directrices
𝑬𝟏, 𝑬𝟐, 𝑬𝟑, 𝑬𝟒= Endpoints of the
Latera Recta
𝒂= distance from the center to
vertex
𝒃= distance from the center to
one endpoint of the conjugate
axis.
𝒄= distance from the center to
Focus.
𝒆= eccentricity
𝟐a= length of the transverse axis
𝟐𝒃= length of the conjugate axis
2c= focal length
27. Exercises
27
Convert the general form to standard form then find the center,
vertices, foci, endpoints of the conjugate axis, endpoints of the latera recta,
Asymptote, eccentricity, length of transverse axis, length of the conjugate
axis, and length of the latus rectum.
1. 9𝑦2
−25𝑥2
+ 200𝑥 − 54𝑦 − 544 = 0