Programmed tutorial: Hyperbola Graph: Determine its Equation: Example 1

Let's determine the equation of this hyperbola. alt="Java applet graph of a hyperbola, foci, vertices." Your browser is completely ignoring the <APPLET> tag!

First write the general equation.

How do we interpret (h, k)?

The point (h, k) is the “middle” of the diagram.

Read the values of h and k from the diagram.

Since the middle is at the origin, (h, k) must be (0, 0), or h = 0 and k = 0.

What do we call the two black dots?

They are the foci (plural).

What is the singular form of this word?

focus.

How are their locations related to a and b?

Indirectly. We define their distance from the “middle” as the distance c.

How is c related to a and b?

c² = a² + b².

Determine c from the diagram.

Since the distance of the foci from the “middle” is 5 units, c = 5.

What do we call the red dots?

They are the vertices (plural).

What is the singular form of this word?

vertex

How are the coordinates of the vertices related to a, b, c, h, k?

Each vertex is a distance a from the “middle” at (h, k).

Determine that distance in the diagram.

Since each vertex is 4 units from the “middle”, we have a = 4.

How do we determine b?

We need the asymptotes to be added to the diagram.

Draw them in a diagram on paper. Then check your graph by clicking “Next”.

alt="Java applet graph of a hyperbola, foci, vertices, asymptotes." Your browser is completely ignoring the <APPLET> tag!

How are the asymptotes related to a and b?

The asymptotes have a slope

Do they intersect the curves?

No, the curves approach the asymptotes.

Determine the slope of the asymptote rising to the right.

The slope is ľ.

Determine b from this.

We have ľ = b/a.
Since we have found that a = 4, we have ľ = b/4.
Consequently, b = 3.

Now that we have values of a, b, c from the diagram, we should check that they agree with c² = a² + b². Do they?

We have a² + b² = 4² + 3² = 25 = 5² = c². Yes they do agree.

Each of the conics has a definition in terms of distances from points or lines.
State the definition of the hyperbola this way.

The hyperbola is the locus of all points for which the difference of the distances from the foci is a constant.

For the blue point, show these distances on paper. Then check by clicking “Next”.

alt="Java applet graph of a hyperbola, foci, vertices, distances." Your browser is completely ignoring the <APPLET> tag!

Do the same for the green point.

alt="Java applet graph of a hyperbola, foci, vertices, distances." Your browser is completely ignoring the <APPLET> tag!

Note that d₄ ą d₂ and d₃ ą d₁.

What is the same for both points?

The difference of the distances is constant:
d₂ – d₁ = d₄ – d₃.

Show that on the diagram.

alt="Java applet graph of a hyperbola, foci, vertices, difference of distances." Your browser is completely ignoring the <APPLET> tag! Here we have drawn d₁ in blue beside d₂, and d₃ beside d₄. The yellow lines represent the differences:
green - blue = yellow, or
d₂ – d₁ and d₄ – d_3.

Are the yellow lines the same length?

Yes, as required by the definition of the hyperbola.

Finally, combine these results to get the equation of the hyperbola.

, becomes

, or

The end. If you found this helpful and would recommend that I create more pages like this one, please let me know: Email to John Taylor