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# Conic Concepts -- Hyperbolas

Module by: Kenny M. Felder. E-mail the author

Summary: This module introduces the definition and formula of a hyperbola, including example problems.

## The Definition of a Hyperbola

A hyperbola is the strangest-looking shape in this section. It looks sort of like two back-to-back parabolas. However, those shapes are not exactly parabolas, and the differences are very important.

Surprisingly, the definition and formula for a hyperbola are very similar to those of an ellipse.

### Definition of a Hyperbola

Take two points. (Each one is a focus; together, they are the foci.) A hyperbola is the set of all points in a plane that have the following property: the distance from the point to one focus, minus the distance from the point to the other focus, is some constant.

The entire definition is identical to the definition of an ellipse, with one critical change: the word “plus” has been changed to “minus.”

One use of hyperbolas comes directly from this definition. Suppose two people hear the same noise, but one hears it ten seconds earlier than the first one. This is roughly enough time for sound to travel 2 miles. So where did the sound originate? Somewhere 2 miles closer to the first observer than the second. This places it somewhere on a hyperbola: the set of all points such that the distance to the second point, minus the distance to the first, is 2.

Another use is astronomical. Suppose a comet is zooming from outer space into our solar system, passing near (but not colliding with) the sun. What path will the comet make? The answer turns out to depend on the comet’s speed.

 If the comet’s speed is low, it will be trapped by the sun’s gravitational pull. The resulting shape will be an elliptical orbit. If the comet’s speed is high, it will escape the sun’s gravitational pull. The resulting shape will be half a hyperbola.

We see in this real life example, as in the definitions, a connection between ellipses and hyperbolas.

## The Formula of an Hyperbola

With hyperbolas, just as with ellipses, it is crucial to start by distinguishing horizontal from vertical. It is also useful to pay close attention to which aspects are the same as ellipses, and which are different.

Table 2: Mathematical Formula for a Hyperbola with its Center at the Origin
Horizontal Vertical
x2a2y2b2=1x2a2y2b2=1 size 12{ { {x rSup { size 8{2} } } over {a rSup { size 8{2} } } } - { {y rSup { size 8{2} } } over {b rSup { size 8{2} } } } =1} {} y2a2x2b2=1y2a2x2b2=1 size 12{ { {y rSup { size 8{2} } } over {a rSup { size 8{2} } } } - { {x rSup { size 8{2} } } over {b rSup { size 8{2} } } } =1} {}

And of course, the usual rules of permutations apply. For instance, if we replace x x with x h xh, the hyperbola moves to the right by h h. So we have the more general form:

Table 3: Formula for a Hyperbola with its Center at xxx(h,k)
Horizontal Vertical
(xh)2a2(yk)2b2=1(xh)2a2(yk)2b2=1 size 12{ { { $$x - h$$ rSup { size 8{2} } } over {a rSup { size 8{2} } } } - { { $$y - k$$ rSup { size 8{2} } } over {b rSup { size 8{2} } } } =1} {} (yk)2a2(xh)2b2=1(yk)2a2(xh)2b2=1 size 12{ { { $$y - k$$ rSup { size 8{2} } } over {a rSup { size 8{2} } } } - { { $$x - h$$ rSup { size 8{2} } } over {b rSup { size 8{2} } } } =1} {}

The key to understanding hyperbolas is understanding the three constants aa, b b, and c c.

Table 4
Horizontal Hyperbola Vertical Hyperbola
Where are the foci? Horizontally around the center Vertically around the center
How far are the foci from the center? cc cc
What is the “transverse axis”? The (horizontal) line from one vertex to the other The (vertical) line from one vertex to the other
How long is the transverse axis? 2 a 2a 2 a 2a
Which is biggest? cc is biggest. c > a c>a, and c > b c>b. cc is biggest. c > a c>a, and c > b c>b.
crucial relationship c 2 = a 2 + b 2 c 2 = a 2 + b 2 c 2 = a 2 + b 2 c 2 = a 2 + b 2

Having trouble keeping it all straight? Let’s make a list of similarities and differences.

### Similarities between Hyperbolas and Ellipses

• The formula is identical, except for the replacement of a+a+ with a-a-.
• The definition of aa is very similar. In a horizontal ellipse, you move horizontally aa from the center to the edges of the ellipse. (This defines the major axis.) In a horizontal hyperbola, you move horizontally aa from the center to the vertices of the hyperbola. (This defines the transverse axis.)
• bb defines a different, perpendicular axis.
• The definition of cc is identical: the distance from center to focus.

### Differences Between Hyperbolas and Ellipses

• The biggest difference is that for an ellipse, a a is always the biggest of the three variables; for a hyperbola, cc is always the biggest. This should be evident from looking at the drawings (the foci are inside an ellipse, outside a hyperbola). However, this difference leads to several other key distinctions.
• For ellipses, a 2 = b 2 + c 2 a 2 = b 2 + c 2 . For hyperbolas, c 2 = a 2 + b 2 c 2 = a 2 + b 2 .
• For ellipses, you tell whether it is horizontal or vertical by looking at which denominator is greater, since aa must always be bigger than bb. For hyperbolas, you tell whether it is horizontal or vertical by looking at which variable has a positive sign, the x 2 x 2 or the y 2 y 2 . The relative sizes of a a and b b do not distinguish horizontal from vertical.

In the example below, note that the process of getting the equation in standard form is identical with hyperbolas and ellipses. The extra last step—rewriting a multiplication by 4 as a division by 1414—can come up with ellipses as easily as with hyperbolas. However, it did not come up in the last example, so it is worth taking note of here.

### Example 1: Putting a Hyperbola in Standard Form

 Graph 3 x 2 – 12 y 2 – 18 x – 24 y + 12 = 0 3 x 2 –12 y 2 –18x–24y+12=0 The problem. We recognize this as a hyperbola because it has an x 2 x 2 and a y 2 y 2 term, and have different signs (one is positive and one negative). 3 x 2 – 18 x - 12 y 2 – 24 y = -12 3 x 2 –18x-12 y 2 –24y=-12 Group together the xx terms and the yy terms, with the number on the other side. 3 ( x 2 – 6 x ) –12 ( y 2 + 2 y ) = -12 3( x 2 –6x)–12( y 2 +2y)=-12 Factor out the coefficients of the squared terms. In the case of the y 2 y 2 for this particular equation, the coefficient is minus 12. 3 ( x 2 – 6 x + 9 ) – 12 ( y 2 + 2 y + 1 ) = -12 + 27 – 12 3( x 2 –6x+9)–12( y 2 +2y+1)=-12+27–12 Complete the square twice. Adding 9 inside the first parentheses adds 27; adding 1 inside the second set subtracts 12. 3 ( x - 3 ) 2 – 12 ( y + 1 ) 2 = 3 3(x-3 ) 2 –12(y+1 ) 2 =3 Rewrite and simplify. ( x - 3 ) 2 – 4 ( y + 1 ) 2 = 1 (x-3 ) 2 –4(y+1 ) 2 =1 Divide by 3, to get a 1 on the right. Note, however, that we are still not in standard form, because of the 4 that is multiplied by ( y + 1 ) 2 (y+1 ) 2 . The standard form has numbers in the denominator, but not in the numerator. ( x - 3 ) 2 - ( y + 1 ) 2 1 4 = 1 (x-3 ) 2 - ( y + 1 ) 2 1 4 =1 Dividing by 1 4 1 4 is the same as multiplying by 4, so this is still the same equation. But now we are in standard form, since the number is on the bottom.

However, the process of graphing a hyperbola is quite different from the process of graphing an ellipse. Even here, however, some similarities lurk beneath the surface.

### Example 2: Graphing a Hyperbola in Standard Form

 Graph ( x - 3 ) 2 – ( y + 1 ) 2 1 4 = 1 (x-3 ) 2 – ( y + 1 ) 2 1 4 =1 The problem, carried over from the example above, now in standard form. Center: (3,–1) Comes straight out of the equation, both signs changed, just as with circles and ellipses. a = 1 a=1 b = 1 2 b= 1 2 The square roots of the denominators, just as with the ellipse. But how do we tell which is which? In the case of a hyperbola, the a a always goes with the positive term. In this case, the x 2 x 2 term is positive, so the term under it is a 2 a 2 . Horizontal hyperbola Again, this is because the x 2 x 2 term is positive. If the y 2 y 2 were the positive term, the hyperbola would be vertical, and the number under the y 2 y 2 term would be considered a 2 a 2 . c = 1 2 + ( 1 2 ) 2 = 5 4 = 5 2 c= 1 2 + ( 1 2 ) 2 = 5 4 = 5 2 Remember that the relationship is different: for hyperbolas, c 2 = a 2 + b 2 c 2 = a 2 + b 2 Now we begin drawing. Begin by drawing the center at (3,–1). Now, since this is a horizontal ellipse, the vertices will be aligned horizontally around the center. Since a = 1 a=1, move 1 to the left and 1 to the right, and draw the vertices there. In the other direction—vertical, in this case—we have something called the “conjugate axis.” Move up and down by b b ( 1 2 1 2 in this case) to draw the endpoints of the conjugate axis. Although not part of the hyperbola, they will help us draw it. Draw a box that goes through the vertices and the endpoints of the conjugate axis. The box is drawn in dotted lines to show that it is not the hyperbola. Draw diagonal lines through the corners of the box—also dotted, because they are also not the hyperbola.These lines are called the asymptotes, and they will guide you in drawing the hyperbola. The further it gets from the vertices, the closer the hyperbola gets to the asymptotes. However, it never crosses them. Now, at last, we are ready to draw the hyperbola. Beginning at the vertices, approach—but do not cross!—the asymptotes. So you see that the asymptotes guide us in setting the width of the hyperbola, performing a similar function to the latus rectum in parabolas.

The hyperbola is the most complicated shape we deal with in this course, with a lot of steps to memorize.

But there is also a very important concept hidden in all that, and that is the concept of an asymptote. Many functions have asymptotes, which you will explore in far greater depth in more advanced courses. An asymptote is a line that a function approaches, but never quite reaches. The asymptotes are the easiest way to confirm that a hyperbola is not actually two back-to-back parabolas. Although one side of a hyperbola resembles a parabola superficially, parabolas do not have asymptotic behavior—the shape is different.

Remember our comet? It flew into the solar system at a high speed, whipped around the sun, and flew away in a hyperbolic orbit. As the comet gets farther away, the sun’s influence becomes less important, and the comet gets closer to its “natural” path—a straight line. In fact, that straight line is the asymptote of the hyperbolic path.

Before we leave hyperbolas, I want to briefly mention a much simpler equation: y = 1 x y= 1 x . This is the equation of a diagonal hyperbola. The asymptotes are the x x and y y axes.

Although the equation looks completely different, the shape is identical to the hyperbolas we have been studying, except that it is rotated 45°.

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