10-4 Hyperbolas
HYPERBOLA
Holt Algebra 2
10-4 Hyperbolas
After going through this module, you are
expected to:
define a hyperbola. (STEM_PC11AGId-1)
determine the standard form of equation of
hyperbola (STEM_PC11AGId-1)
- from a graph
- from given parts
Holt Algebra 2
10-4 Hyperbolas
Holt Algebra 2
10-4 Hyperbolas
Applications
Holt Algebra 2
10-4 Hyperbolas
A hyperbola is a set of points P(x, y) in a plane such
that the difference of the distances from P to fixed
points F1 and F2, the foci, is constant. For a hyperbola,
d = |PF1 – PF2 |, where d is the constant difference.
You can use the distance formula to find the equation
of a hyperbola.
Holt Algebra 2
10-4 Hyperbolas
Parts of a Hyperbola
Holt Algebra 2
10-4 Hyperbolas
Holt Algebra 2
10-4 Hyperbolas
Derivation of the formula
Holt Algebra 2
10-4 Hyperbolas
Holt Algebra 2
10-4 Hyperbolas
Derivation of the formula
Holt Algebra 2
10-4 Hyperbolas
Holt Algebra 2
10-4 Hyperbolas
Holt Algebra 2
10-4 Hyperbolas
Holt Algebra 2
10-4 Hyperbolas
Holt Algebra 2
10-4 Hyperbolas
Holt Algebra 2
10-4 Hyperbolas
Holt Algebra 2
10-4 Hyperbolas
Holt Algebra 2
10-4 Hyperbolas
Example 3: Find the standard form of the equation of the
hyperbola given foci at (1,-2) and (5,2) while
vertices at (0,2) and (4,2)
Holt Algebra 2
10-4 Hyperbolas
Holt Algebra 2
10-4 Hyperbolas
Source:
https://www.toppr.com/guides/maths/conic-sections/equationofhyperbola/#:~:text=Derivation%20of%20the%20Equation&text=
In%20hyperbola%20c%20%3E%20a%3B%20and,%2Fa)x%20beco
mes%20negative.&text=Therefore%2C%20PF1%20%E2%80%93
%20PF2,a)%20%2B%20a%20%3D%202a.&text=In%20this%20cas
e%2C%20PF1%20%E2%80%93%20PF2%20%3D%202a.
Holt Algebra 2
10-4 Hyperbolas
The standard form of the equation of a hyperbola
depends on whether the hyperbola’s transverse axis is
horizontal or vertical.
Holt Algebra 2
10-4 Hyperbolas
Example 1
Find the vertices, co-vertices, and asymptotes of
each hyperbola, and then graph
2
x2 – y = 1
9
49
Step 1 The vertices are –7, 0) and
(7, 0) and the co-vertices
are (0, –3) and (0, 3).
Step 2 Draw a box using the vertices
and co-vertices. Draw the
asymptotes through the
corners of the box.
Step 3 Draw the hyperbola by
using the vertices and
the asymptotes.
Holt Algebra 2
10-4 Hyperbolas
Example 2: Graphing a Hyperbola
Find the vertices, co-vertices, and asymptotes of each
hyperbola, and then graph.
(x – 3)2
(y + 5)2
–
=1
9
49
Step 1 Center is (3, -5), the vertices are (0, –5)
and (6, –5) and the co-vertices are (3, –12)
and (3, 2) .
Step 2 The asymptotes cross at (3, -5) and have a
slope =
Holt Algebra 2
7
3
10-4 Hyperbolas
Example 2B Continued
Step 3 Draw a box by using
the vertices and covertices. Draw the
asymptotes through
the corners of the box.
Step 4 Draw the hyperbola
by using the vertices
and the asymptotes.
Holt Algebra 2
10-4 Hyperbolas
Example 3A: Writing Equations of Hyperbolas
Write an equation in standard
form for each hyperbola.
Step 1: The graph opens horizontally, so the equation
will be in the form of
x2
a2
–
y2
2 = 1
b
Step 2: Because a = 6 and b = 6, the equation of the
graph is
x2
36
Holt Algebra 2
y2
–
= 1.
36
10-4 Hyperbolas
Example 3B: Writing Equations of Hyperbolas
Write an equation for the hyperbola with center
at the origin, vertex (4, 0), and focus (10, 0).
Step 1 Because the vertex and the focus are on the
horizontal axis
x2
16
y2
–
b2
=1
Step 2 Use focus2 = 16 + other denominator.
x2
Step 3 The equation of the hyperbola is
16
Holt Algebra 2
y2
–
=1 .
84
10-4 Hyperbolas
A.
1. Graph the ellipses
Notes
x2 +
4
y2
=1
9
.
2
(x-3)2 + (y+1) = 1
B.
25
144
2. Graph the hyperbola
2
(x-3)2 - (y+1) . = 1
25
144
3. Find the vertices, co-vertices, and asymptotes
of
, then graph.
4. Write an equation in standard form for a hyperbola with
center hyperbola (4, 0), vertex (10, 0), and focus (12, 0).
Holt Algebra 2
10-4 Hyperbolas
Notes
3. Find the vertices, co-vertices, and asymptotes
of
, then graph.
vertices: (–6, ±5); co-vertices (6, 0), (–18, 0);
5
(x + 6)
asymptotes: y = ±
12
Holt Algebra 2
10-4 Hyperbolas
Notes:
4. Write an equation in standard form for a
hyperbola with center hyperbola (4, 0), vertex
(10, 0), and focus (12, 0).
Holt Algebra 2
10-4 Hyperbolas
The values a, b, and c, are related by the equation
c2 = a2 + b2. Also note that the length of the transverse axis is 2a and the length of the conjugate is 2b.
Holt Algebra 2
10-4 Hyperbolas
As with circles and ellipses, hyperbolas do not have
to be centered at the origin.
Holt Algebra 2
10-4 Hyperbolas
Hyperbolas: Extra Info
The following power-point slides contain
extra examples and information.
Reminder: Lesson Objectives
Write the standard equation for a hyperbola.
Graph a hyperbola, and identify its center, vertices,
co-vertices, foci, and asymptotes.
Holt Algebra 2
10-4 Hyperbolas
Notice that as the parameters change, the graph
of the hyperbola is transformed.
Holt Algebra 2
10-4 Hyperbolas
Check It Out! Example 3a: Graphing
Find the vertices, co-vertices, and asymptotes of
each hyperbola, and then graph.
2
x2 – y = 1
36
16
2
x
Step 1 The equation is in the form 2
a
–
y2
so
2 = 1
b
the transverse axis is horizontal with center
(0, 0).
Holt Algebra 2
10-4 Hyperbolas
Check It Out! Example 3a Continued
Step 2 Because a = 4 and b = 6, the vertices are
(4, 0) and (–4, 0) and the co-vertices are
(0, 6) and . (0, –6).
Step 3 The equations of the asymptotes are
y=
Holt Algebra 2
3
2
x
and y = –
3
x .
2
10-4 Hyperbolas
Check It Out! Example 3
Step 4 Draw a box by using
the vertices and covertices. Draw the
asymptotes through
the corners of the box.
Step 5 Draw the hyperbola
by using the vertices
and the asymptotes.
Holt Algebra 2
10-4 Hyperbolas
Check It Out! Example 3b: Graphing
Find the vertices, co-vertices, and asymptotes of
each hyperbola, and then graph.
(y + 5)2
(x – 1)2
–
=1
1
9
Step 1 The equation is in the form
(y – k)2
(x – h)2
–
= 1 so the
2
2
a
b
transverse axis is vertical with
center (1, –5).
Holt Algebra 2
10-4 Hyperbolas
Check It Out! Example 3b Continued
Step 2 Because a = 1 and b =3, the vertices are
(1, –4) and (1, –6) and the co-vertices
are (4, –5) and (–2, –5).
Step 3 The equations of the asymptotes are
y+5=
Holt Algebra 2
1
1
(x – 1) and y + 5 = –
3
3
(x – 3).
10-4 Hyperbolas
Check It Out! Continued
Step 4 Draw a box by using
the vertices and covertices. Draw the
asymptotes through
the corners of the box.
Step 5 Draw the hyperbola
by using the vertices
and the asymptotes.
Holt Algebra 2
10-4 Hyperbolas
Check It Out! Example 2a: Writing Equations
Write an equation in standard form for each
hyperbola.
Vertex (0, 9), co-vertex (7, 0)
Step 1 Because the vertex is on the vertical
axis, the transverse axis is vertical and the
2
2
x
y
equation is in the form 2 –
.
2 = 1
a
Step 2 a = 9 and b = 7.
b
Step 3 Write the equation.
Because a = 9 and b = 7, the equation of the
graph is
Holt Algebra 2
y2
92
–
2
x2
y
, or
2 = 1
7
81
x2
–
= 1.
49
10-4 Hyperbolas
Check It Out! Example 2b: Writing Equations
Write an equation in standard form for each
hyperbola.
Vertex (8, 0), focus (10, 0)
Step 1 Because the vertex and the focus are on
the horizontal axis, the transverse axis is
horizontal and the equation is in the form
x2
a2
Holt Algebra 2
–
y2
2 = 1.
b
10-4 Hyperbolas
Check It Out! Example 2b Continued
Step 2 a = 8 and c = 10; Use c2 = a2 + b2 to solve
for b2.
102 = 82 + b2 Substitute 10 for c, and 8 for a.
36 = b2
Step 3 The equation of the hyperbola is
Holt Algebra 2
x2
64
y2
–
= 1.
36