The equation and properties of a hyperbola are explored interactively using an applet. The equation used has the form
x^{ 2} / a^{ 2} - y ^{ 2} / b^{ 2} = 1
where a and b are positive real numbers.
The exploration is carried out by changing the parameters a and b included in the above equation. Similar interactive tutorials on the equation of ellipse , parabola and circle can be found in this site.
Also a tutorial on finding properties of hyperbolas analytically can be found in this site.
Interactive Tutorial
click on the button above "click here to start" and MAXIMIZE the window obtained.
When the applet is started each of the parameters a and b in the equation of the hyperbola shown above is equal to 1. If for some reasons they are not, use the sliders top/left to set each one of them to 1.
In the main panel, a hyperbola is plotted. Note the following: F and F' are the foci (plural of focus); V and V' are the vertices of the hyperbola. In the main menu, top/left, d1 and d2 are the distances from F to M and from F' to M
respectively.
d1 = distance from F to M
d2 = distance from F' to M
where point M is a marker that can be positioned anywhere by clicking on the position desired.
Explore the definition of the hyperbola
Click anywhere on the graph of the hyperbola (blue), adjust point M so that it is on the graph. Read distances d1 and d2 (top/left) and find the absolute value of their difference: | d1 - d2 |. Repeat this experiment several times. Show that this difference is constant (approximately). Define the set of points that make a hyperbola.
Foci
Set parameters a to 1 and b to 1. Click on F to position M on F then read the coordinates of M (top/left): M(1.4 , 0). These are the coordinates of F of the form (c , 0). Verify that
c = √(a^{2} + b^{2})
Click on F' and verify that F' has coordinates (-c , 0). Repeat this last experiment for several values of a and b.
Vertices
V and V' are the x intercepts of the graph of the parabola. Show analytically that V and V' has coordinates (a , 0) and (-a , 0) respectively. Check this results graphically by reading the
coordinates of V and V'. (set a to values such as 1.0, 2.0 ...).
Asymptotes
The asymptotes are the two red broken lines. What are they?
Rewrite the equation of the hyperbola so that the term in y is to the left and all other terms to the right.
y^{2} / b^{2} = x^{2} /a^{2}
- 1
as | x | becomes very large the right term is dominated by the term
x^{2} / a^{2}
and the whole equation of the hyperbola may be approximated by:
y^{2} / b^{2 } = x^{2} / a^{2}
The above equation may be written as two separate equations(solving for y).
y = (b/a) x
y = - (b/a) x
So when | x | is very large ( x very large or x very small), the graph of the parabola behaves as the graph of the lines y = (b/a) x and y = - (b/a) x which are called asymptotes.
When graphing hyperbolas, it is easier to draw a rectangle (shown in red) of length 2a (length of the transverse axis) and width 2b (length of the conjugate axis) and the asymptotes are the extensions of the diagonals of the
rectangles as shown in the main panel of the applet.
Exercises
Given the following equation of the hyperbola
x^{2}/4 - y ^{2}/9 = 1
a) Compare the given equation to the standard above and find a and b.
b) Find the coordinates of the foci.
c) Find the x intercepts of the graph of the given equation.
d) Find the equations of the asymptotes.
e) Use the applet to check the answers to parts b, c and d above.