Hyperbola

hyperbola focus on horizontal transverse axis.

1. Hyperbola
Helen M. Ogoc
Punta National High School
Punta, Dipolog City
With Teacher

2. Martin-Gay, Developmental Mathematics 2
Hyperbola
Definition:
A hyperbola is the set of all points P such that the
difference of the distance from P to two fixed points,
called the foci, is constant.
The line through the foci intersects the hyperbola at two
points, the vertices.
The line segment joining the vertices is the transverse
axis, and its midpoint is the center of the hyperbola.
A hyperbola has two branches and two asymptotes.

3. Martin-Gay, Developmental Mathematics 3
The asymptotes contain the diagonals of a
rectangle centered at the hyperbola’s center.
To get the correct shape of the hyperbola, we need
to find the asymptotes of the hyperbola.
The asymptotes are lines that are approached but
not touched or crossed.
These asymptotes are boundaries of the hyperbola.
This is one difference between a hyperbola and a
parabola.

4. Martin-Gay, Developmental Mathematics 4
For the hyperbolas that
open right/left,
the asymptotes are:
and for hyperbolas
opening up/down, the
asymptotes are:

5. Martin-Gay, Developmental Mathematics 5
Characteristics of Hyperbola (center of origin)
Equation Transverse AxisAsymptotes Vertices

Horizontal Y = ±(b/a)x (±a, 0)

Vertical Y = ±(a/b)x (0, ±a)
Foci: c² = a² + b²

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Diagram of Hyperbola Standard Form

7. Martin-Gay, Developmental Mathematics 7
Diagram of Hyperbola Standard Form

8. Translation of
Hyperbola

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If the hyperbola opens right/left the translation is:
with the equations of the asymptotic lines as:
y - k = + (b/a)(x - h)
If the hyperbola opens up/down the translation is:
with the equations of the asymptotic lines as:
y - k = + (a/b)(x - h)

10. horizontal transverse axis
Hyperbola

11. Martin-Gay, Developmental Mathematics 11
Hyperbola horizontal transverse axis
"a" is the number in the denominator of the positive term
If the x-term is positive, then the hyperbola is horizontal
a = semi-transverse axis
b = semi-conjugate axis
center: (h, k)
vertices: (h + a, k), (h - a, k)

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c = distance from the center to each focus along

the transverse axis
foci: (h + c, k), (h - c, k)
The eccentricity e > 1
The eccentricity e > 1
Hyperbola horizontal transverse axis

13. Martin-Gay, Developmental Mathematics 13
Hyperbola horizontal transverse axis

14. Martin-Gay, Developmental Mathematics 14
Hyperbola horizontal transverse axis
Draw the hyperbola given by 
Solution:
Therefore, a = 3, b = 2 and c = 3.6.
This hyperbola opens right/left because it is in the form x - y.
a
2
= 9, b
2
= 4, c
2
= 9 + 4 = 13.

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Hyperbola horizontal transverse axis






















a
b

16. Martin-Gay, Developmental Mathematics 16 e!s "
Example 2:
Hyperbola horizontal transverse axis

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Solution:
From the standard form:
Hyperbola horizontal transverse axis
x - h x - 2 h = 2
y - k y - 0 k = 0
Center: (2,0)
a
2
= 4 a = 2b
2
= 25b = 5

This means we move 2 units to the left and right of the
center and 5 units up and down from the center to arrive at
points on the guide rectangle.
The asymptotes pass through the center of the hyperbola as
well as the corners of the rectangle.

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Hyperbola horizontal transverse axis

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y
2
term is being subtracted from the x
2
term,
the branches of the hyperbola open to the left and right
this means that the transverse axis lies along the x-axis
the conjugate axis lies along the vertical line x = 2
the vertices of the hyperbola are where the hyperbola
intersects the transverse axis
the vertices are 2 units to the left and right of (2, 0)
which is the center
vertices: (0,0) and (4, 0)
Hyperbola horizontal transverse axis

20. Martin-Gay, Developmental Mathematics 20
To find the foci, we need
(-3.39,0) and (7.39,0)
To determine the equations of the asymptotes, recall that the
asymptotes go through the center of the hyperbola, (2; 0), as
well as the corners of guide rectangle,
Using the point-slope equation of a line,
We get,
Hyperbola horizontal transverse axis

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Putting it all together, we get
Center: (2,0)
vertices: (0; 0) and (4; 0)
(-3.39,0) and (7.39,0)
Asymptotes:
Hyperbola horizontal transverse axis

22. Martin-Gay, Developmental Mathematics 22)v"o
Exercises :
1. Graph the hyperbola. Find the center, the lines which
contain the transverse and conjugate axes, the vertices, the
foci and the equations of the asymptotes of
Center:
Tranverse Axis:
Congugate Axis:
Vertices:
Foci:
Asymptote:
(4, 2)
y = 2
x = 4
Hyperbola horizontal transverse axis

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Hyperbola horizontal transverse axis

24. Martin-Gay, Developmental Mathematics 24)v"o
Exercises :
Center:
Tranverse Axis:
Congugate Axis:
Vertices:
Foci:
Asymptote:
(3,-1)
y = -1
x = 3
Hyperbola horizontal transverse axis
2. Put the equation in standard form. Find the center, the lines
which contain the transverse and conjugate axes, the vertices,
the foci and the equations of the asymptotes.
9x
2
- 25y
2
- 54x - 50y - 169 = 0
Equation in standard form:

25. Martin-Gay, Developmental Mathematics 25icsvos
Exercises :
Hyperbola horizontal transverse axis
3. Find the standard form of the equation of the hyperbola
which has the given properties..
a. Vertex (0, 1), Vertex (8, 1), Focus (-3, 1)
c. Vertices (3, 2), (13, 2);
Endpoints of the Conjugate Axis (8, 4), (8, 0)

26. Martin-Gay, Developmental Mathematics 26psv
Answers :
Hyperbola horizontal transverse axis
a. Vertex (0, 1), Vertex (8, 1), Focus (-3, 1)
c. Vertices (3, 2), (13, 2);
Endpoints of the Conjugate Axis (8, 4), (8, 0)

27. Martin-Gay, Developmental Mathematics 27
Hyperbola horizontal transverse axis
4. Find the standard form of the following equation using
then graph the conic section.
d. 9x
2
- 4y
2
- 36x - 24y - 36 = 0
a. x
2
- 2x - 4y - 11 = 0
b. x
2
+ y
2
- 8x + 4y + 11 = 0
c. 9x
2
+ 4y
2
- 36x + 24y + 36 = 0

28. Martin-Gay, Developmental Mathematics 28
Hyperbola horizontal transverse axis
4. Find the standard form of the following equation using
then graph the conic section.
a. x
2
- 2x - 4y - 11 = 0 b. x
2
+ y
2
- 8x + 4y + 11 = 0psv
Answers :

29. Martin-Gay, Developmental Mathematics 29
Hyperbola horizontal transverse axis
4. Find the standard form of the following equation using
then graph the conic section.
d. 9x
2
- 4y
2
- 36x - 24y - 36 = 0c. 9x
2
+ 4y
2
- 36x + 24y + 36 = 0

30. Martin-Gay, Developmental Mathematics 30
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