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### Course: Precalculus (Eureka Math/EngageNY)>Unit 3

Lesson 5: Topic A: Lesson 8: Hyperbolas

# Foci of a hyperbola from equation

Sal discusses the foci of hyperbolas and shows how they relate to hyperbola equations. Created by Sal Khan.

## Want to join the conversation?

• So a hyperbola is essentially an "inside out" ellipse?
• It is technically the opposite of a ellipse. One of the x/y is negative making it "inside out."
• I am confused. Where is the point 'b' on hyperbola?
• "b" is not a point on a hyperbola, but the number is important to the overall shape. The b comes in when finding the slope of asymptotes of the hyperbola.

"(b/a) x" will give the equations for those lines, in the event the it is centered on the origin.

"(b/a) (x-c) + d", where c is the change in x and d is the change in y, will solve for any hyperbola of the origin.
• can the focal lengths of an ellipse be any distance away from the center along the major axis ??
• Focal length is the distance away from the center the 2 Foci are. Foci will always exist on the major radius so no.
(1 vote)
• I'm confused of the definition of ellipse and hyperbola.. Why is this video called 'Foci of hyperbola' while what being talked about is the foci of ellipse? Or is there any close relationship between ellipse and hyperbola? Many thanks!
• At the beginning of the video he shows you the ellipse because he wanted you to see that f = sqrt(a^2 - b^2) is an equation that applies to the ellipse and then then after that he starts to talk about the hyperbola and how the equation is f = sqrt(a^2 + b^2), which shows the relationship you are asking of between the ellipse and the hyperbola because you can see the only change is that the - became a +
• Does anyone hear the alarm going off in the background?
• Shouldn't the focal length just be the square root of the absolute value of a squared minus b squared? This way it doesn't matter which is bigger.
f=sqrt(|a^2-b^2|)
• ??? sqrt(abs(3^2-4^2))=sqrt(5)
sqrt(abs(4^2+3^2))=sqrt(25)=5.

If you are just talking about an ellipse then yes that is right.
• Does Sal have any proof videos for d1 + d2 = 2a for ellipses or |d2 - d1| = 2a for hyperbolas?
• I do not understand why the asymptote is 4/3? Is it not that x^2= (9/16)*y^2, so the asymptote is x=(3/4)y
• what does the ' a ' and ' b ' represent ? a radius of something ?
(1 vote)
• 'a' is the distance between the center of the hyperbola and the vertices. I am not sure if 'b' represents a specific distance related to the hyperbola, but as we see in this video, it is related to the focus, and it is also related to the slope of the asymptotes: b / a

My suspicion is that an equation that uses only distances defined by the hyperbola might substitute |a^2 - f^2| for b^2, but then you wouldn't have easy access to the slope of the asymptotes, which would be inconvenient.
• I don't get why the formula for a hyperbola is so similar to the formula for an ellipse. Can someone explain? Thanks.
Also, at wasn't sal using the formula of an ellipse and using the focal length formula of a hyperbola?
• They are similar because the equation for a hyperbola is the same as an ellipse except the equation for a hyperbola has a - instead of a + (in the graphical equation). As for your second question, Sal is using the foci formula of the hyperbola, not an ellipse. The foci formula for an ellipse is

c^2=|a^2-b^2|

And the foci formula for a hyperbola is:

c^2 = a^2 + b^2 (same as the pythagorean theorem)

does this help?
(1 vote)