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### Course: Algebra (all content)>Unit 17

Lesson 8: Introduction to hyperbolas

# Intro to hyperbolas

Sal introduces the standard equation for hyperbolas, and how it can be used in order to determine the direction of the hyperbola and its vertices. Created by Sal Khan.

## Want to join the conversation?

• At , Sal got rid of the -b^2, how is that possible?
• As `x` approaches infinity, the `b²` term becomes less and less significant.
For example, consider `x = 1,000,000` and `b = 5` .
• I have actually a very basic question. The variables a and b, do they have any specific meaning on the function or are they just some paramters?
• Yes, they do have a meaning, but it isn't specific to one thing. You may need to know them depending on what you are being taught. If you are learning the foci (plural of focus) of a hyperbola, then you need to know the Pythagorean Theorem:
a^2 + b^2 = c^2
The foci are +-c
Even if you aren't learning the foci, you still need them for the asymptotes.
• Is a parabola half an ellipse? Also can the two "parts" of a hyperbola be put together to form an ellipse?
• They look a little bit similar, don't they? But no, they are three different types of curves.
• At how does it become b squared over a squared x squared?
• For anyone else caught on this:

b^2 * x^2/a^2 is the same thing as b^2/a^2 * x^2/1. Which you can then rewrite as b^2/a^2 * x^2.
(1 vote)
• Well what'll happen if the eccentricity of the hyperbolic curve is equal to infinity?
Will it be equal to a pair of two parallel straight lines?
copyd this question quz cant see answers!
• its a bit late, but an eccentricity of infinity forms a straight line. This is because eccentricity measures who much a curve deviates from perfect circle. So circle has eccentricity of 0 and the line has infinite eccentricity.
• Hang on a minute why are conic sections called conic sections.
(1 vote)
• At about Sal states (in purple) that for x²=a² x = +/- sqrt(a). I don't get this? If you square root both sides you get x=a, right?
• This is a mistake in the video. I'm not sure why it has not been corrected. (Thanks for the time-stamp: they really help with a question like this :-). Notice that Sal doesn't use this mistaken result: when he locates his solutions on the x-axis, immediately after making his mistake, he labels them, correctly, as "a" and "-a"
• circle equation is related to radius.how to hyperbola equation ?
• 2y=-5x-30
y=-5x/2-15
• at about , won't the second line's slope be -a/b instead of -b/a? i learned that perpendicular lines were negative reciprocals, not just negative opposites.
• the asymptotes are not perpendicular to each other.
• What is the intuition behind having a negative sign in hyperbola equations X^2/a^2 - Y^2/b^2 = 1. This is the only difference between an ellipse and a hyperbola equation. Can anyone please explain me the reason of the sign intuition changing the shape of a ellipse curve to a hyperbola ?
• As I'm learning/reviewing this myself, I would take this answer with a grain of salt.

The shape of a hyperbola, as opposed to an ellipse, has to do with the behaviors of x and y as they approach infinity. A hyperbola's equation will result in asymptotes reflected across the x and y axis, while the ellipse's equation will not.

In order to understand why, let's have an equation of a hyperbola and an ellipse, respectively: x^2/9 - y^2/4 = 1; x^2/9 + y^2/4 = 1.

When solving for values of y for the hyperbola, we first rearrange its equation to isolate y:
-y^2/4 = 1 - x^2/9
y^2 = 4/9x^2 - 4
y = +/- the square root of 4/9x^2 - 4 (x can be as large or as negative as you want and still output a real solution).

As x approaches +/- infinity, y approaches +/- 2/3x. If the denominators were equal, then y would ~x, but never reach it, resulting in the asymptotes.

Using the same steps for the ellipse equation, y is +/- 2 - 2/3x, or +/- the square root of 4 - 4/9x^2. Here, x cannot be greater than 3 or less than -3, since that would output an imaginary number.