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1. HYPERBOLA
A Conic Section

2. HYPERBOLA
A hyperbola, like an ellipse, is determined by a
constant and two fixed points, each called a focus point.
However, instead of the sum of two distances being a
constant, as with an ellipse, in the case of the hyperbola, the
difference between two distances is constant.
More specifically, the hyperbola determined by a pair
of foci and a given constant is the set of points (x, y) in the
plane such that the absolute value of the difference between
the distances from (x, y) to the foci is equal to the constant.

3. .
PARTS OF HYPERBOLA
There are six parts of hyperbola, namely, center, foci, vertices, transverse axis,
conjugate axis, asymptotes and latus recta. An auxiliary rectangle is also an
additional part that will help in sketching an accurate asymptotes of the
hyperbola.

6. Purpose
The following properties of the graphs on which we concentrate on our
investigation:
a. Equations of the asymptotes of the graphs of the hyperbolas
b. Distance of each vertex from the origin
c. Coordinates of the vertices
d. Length of the transverse axis
e. Coordinates of the foci
f. Shapes of the graphs

7. Horizontal Transverse Axis Hyperbola
with Center at the Origin Whose a
and b are equal
𝑥2
9
-
𝑦2
9
= 1
𝑎2= 9
𝑎 = 3
𝑏2= 9
𝑏 = 3
𝑐 = 𝑎2 + 𝑏2 Asymptotes
= 18 𝑦 =
3
3
𝑥
= 4.24 𝑦 = −
3
3
𝑥
Length of Transverse Axis: 2(a)
2(3)
Length of Conjugate Axis: 2(b)
2(3)

8. Horizontal Transverse Axis Hyperbola
with Center at the Origin Whose a
and b are equal

9. a. Equation:
𝑥2
𝑎2 -
𝑦2
𝑏2
To get the Asymptotes: 𝑦 =
𝑏
𝑎
𝑥 𝑦 = −
𝑏
𝑎
𝑥
𝑦 =
3
3
𝑥 𝑦 = −
3
3
𝑥
b.
2𝑎
2
=
2(3)
2
𝑎 = 3
Let’s denote that 𝑎 represents the distance of each
vertices from the center.
c. 𝑎, 0 , (−𝑎, 0)
The value of 𝑎 is 3.
3,0 , (−3,0)

10. d. 2𝑎 𝑢𝑛𝑖𝑡𝑠
2 3 = 6 𝑢𝑛𝑖𝑡𝑠
e. 𝑐, 0 , (−𝑐, 0)
The value 𝑐 is 4.24
f.

11. 𝑦2
81
−
𝑥2
81
= 1 Asymptotes
𝑦 =
9
9
𝑥
𝑎2= 81 𝑦 = −
9
9
𝑥
𝑎 = 9
𝑏2= 81
𝑏 = 9
𝑐 = 𝑎2 + 𝑏2
= 162
c = 12.7
Vertical Transverse Axis Hyperbola
with Center at the Origin Whose a and
b are equal

12. Vertical Transverse Axis Hyperbola
with Center at the Origin Whose a and
b are equal

13. a. Equation:
𝑦2
𝑎2 -
𝑥2
𝑏2
To get the Asymptotes: 𝑦 =
𝑎
𝑏
𝑥 𝑦 = −
𝑎
𝑏
𝑥
𝑦 =
9
9
𝑥 𝑦 = −
9
9
𝑥
b.
2𝑎
2
=
2(9)
2
𝑎 = 9
Let’s denote that 𝑎 represents the distance of each vertices from the center.
c. 0, 𝑎 , (0, −𝑎)
The value of 𝑎 is 9.
0,9 , (0, −9) f.
d. 2𝑎 𝑢𝑛𝑖𝑡𝑠
2 9 = 18 𝑢𝑛𝑖𝑡𝑠
e. 0, 𝑐 , (0, −𝑐)
The value 𝑐 is 12.72