LC.01.5 - The Hyperbola
MCR3U - Santowski
1
(A) Hyperbola as Loci

A hyperbola is defined as the set of points such that the difference of the
distances from any point on the hyperbola to two stationary points (called
the foci) is a constant

We will explore the hyperbola from a locus definition in two ways

ex 3. Using grid paper with 2 sets of concentric circles, we can define the
two circle centers as fixed points and then label all other points (P), that
meet the requirement that the difference of the distances from the point (P)
on the hyperbola to the two fixed centers (which we will call foci) will be a
constant i.e. | PF1 - PF2 | = constant. We will work with the example that
PF1 - PF2 = 4 units.

ex 4. Using the GSP program, we will geometrically construct a set of
points that satisfy the condition that | PF1 - PF2 | = constant by following
the directions on the handout
2
(B) Hyperbolas as Loci

3
(C) Hyperbolas as Loci - Algebra

We will now tie in our knowledge of algebra to
come up with an algebraic description of the
hyperbola by making use of the relationship that
| PF1 - PF2| = constant

ex 5. Find the equation of the hyperbola
whose foci are at (+3,0) and the constant
(which is called the difference of the focal radi)
is 4. Then sketch the hyperbola by finding the x
and y intercepts.
4
(C) Hyperbolas as Loci - Algebra

Since we are dealing with distances, we set up our
equation using the general point P(x,y), F1 at (-3,0) and
F2 at (3,0) and the algebra follows on the next slide |PF1
- PF2| = 4
5
(C) Algebraic Work
PF1  PF2  4




 x  32
 y2 


 



 x  3
 y 


2
 x  32
2

4 

 y2 
4

 x  3
2
 y 


2
2
 x  32  y 2   x  32  y 2
2
 1 6  8  x  3  y 2  x 2  6 x  9 
 y2  16  8
x2  6x  9  y2
8
2
 x  32
 x  32
y2
 y 2  1 6  1 2x
2
2
2

 2  x  3  y 2 
  3 x  4 


4 x 2  6 x  9  y 2  1 6  2 4x  9 x 2


4 x 2  9 x 2  2 4x  2 4x  4 y 2  1 6  3 6
 5 x 2  4 y 2  2 0
5x 2  4 y 2  2 0
 5x 2

 20

  4 y2 
 x2   y2 
 y 
 x
















  20 
 4   5 
2
5




 


 

2
2
1
6
(D) Graph of the Hyperbola

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(E) Analysis of the Hyperbola

The equation is (x/2)2 - (y/5)2 = 1 OR 5x2 - 4y2 = 20
The x-intercepts occur at (+2,0) and there are NO y-intercepts
The domain is {x E R | -2 > x > 2} and the range is {y E R }

NOTE that this is NOT a function, but rather a relation

NOTE the relationship between the equation and the intercepts,
domain and range  so to generalize, if the hyperbola has the
standard form equation (x/a)2 - (y/b)2 = 1, then the x-intercepts
occur at (+a,0), and the domain is {x E R | -a > x > a} and the
range is any real number
We can rewrite the equation in the form of (bx)2 - (ay)2 = (ab)2





Note that if the equation is (x/a)2 - (y/b)2 = 1, then the hyperbola
opens along the x-axis
BUT if the equation is (y/a)2 - (x/b)2 = 1 (OR (x/b)2 - (y/a)2 = -1)
then the hyperbola opens along the y-axis
8
(E) Analysis of the Hyperbola

This hyperbola has the form (x/b)2 – (y/a)2 = -1 (or
(y/a)2 – (x/b)2 = 1
9
(E) Analysis of the Hyperbola



To get the correct shape of the hyperbola, we need to find and
graph the asymptotes of the hyperbola  the fact that a hyperbolic
curve approaches asymptotes is what makes it different from a
parabola
For hyperbolas that open along the x-axis, the equation of the
asymptotic lines are y = +(b/a)x
To easily graph the asymptotes on the graph, we simply form a
rectangular box using the values of a and b (corners of the box are
(a,b), (-a,b), (-a,-b), (a,-b))
10
(E) Analysis of the Hyperbola





The axis upon which the hyperbola opens is called the
major axis and lies between the 2 x-intercepts. Its
length is 2a (vice versa if direction of opening is along
y-axis)
The two end points of the major axis (in this case the xintercepts) are called vertices (at (+a,0))
The two foci lie on the major axis
The domain is {x E R | -a > x > a} and the range is any
real number
The asymptotes are at y = + (b/a)x (BUT if the
hyperbola opens up/down, then the equation of the
asymptotes is y = + (a/b)x)
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(F) In-class Examples

Determine the equation of the hyperbola and
then sketch it, labelling the key features, if the
foci are at (0,+5) and the difference of the focal
radii is 6 units (i.e.  the fancy name for the
constant distance sum PF1 - PF2)
The equation you generate should be either
 y2/9 - x2/16 = 1
 or x2/16 – y2/9 = -1

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(G) Internet Links
http://www.analyzemath.com/EquationHyperbol
a/EquationHyperbola.html - an interactive applet
fom AnalyzeMath
 http://home.alltel.net/okrebs/page63.html Examples and explanations from OJK's
Precalculus Study Page
 http://tutorial.math.lamar.edu/AllBrowsers/1314/
Hyperbolas.asp - Ellipses from Paul Dawkins at
Lamar University
 http://www.webmath.com/hyperbolas.html Graphs of ellipses from WebMath.com

13
(G) Homework

AW, p481, Q7ab, 8ab

Nelson text, p616, Q7,15,16,6
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