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Course: Precalculus > Unit 5
Lesson 6: Hyperbolas not centered at the originConic sections: FAQ
What are conic sections?
Conic sections are the shapes you get when you slice a cone at different angles. There are four types of conic sections: ellipses, hyperbolas, parabolas, and circles.
What's the difference between an ellipse and a hyperbola?
Both shapes are conic sections, but they differ in their geometry. An ellipse is a closed curve, shaped like an oval. A hyperbola consists of two separate curves (called "branches") that open away from each other.
What are the key parts of an ellipse?
An ellipse has a few key parts:
- major axis: This is the longest diameter of the ellipse, and it runs through the center of the shape.
- minor axis: This is the shortest diameter of the ellipse, also running through the center of the shape, perpendicular to the major axis.
- foci: These are two points on the major axis that define the shape of the ellipse. The sum of the distances from the foci to any point on the ellipse is always constant.
- center: This is the midpoint of both the major and minor axes.
- vertices: These are the two points on the major axis that are furthest from the center.
- co-vertices: These are the two points on the minor axis that are furthest from the center.
What are the key parts of a hyperbola?
- branches: A hyperbola is made up of two distinct branches or curves that extend away from each other.
- center: The point equidistant from the two branches, around which the hyperbola is symmetrical.
- transverse axis: The line segment connecting the two closest points on the two branches.
- conjugate axis: The line segment perpendicular to the transverse axis, passing through the center.
- foci: The two fixed points located inside each curve of the hyperbola such that, for any point on the hyperbola, the difference of distances to the foci is a constant.
- asymptotes: The two lines that the branches of the hyperbola will approach as they extend further and further away from the center. These asymptotes intersect at the center of the hyperbola.
Where are conic sections used in the real world?
Conic sections show up in a lot of places! For example, the orbits of planets around the sun are elliptical. Hyperbolas are often used in the design of telescopes and antennas. Parabolas are important in physics, as they describe the shape of projectiles in flight.
Want to join the conversation?
I have got three questions:
1. how to calculate the area and the circumference of an
ellipse?
2. Do we need pi to calculate the are and circumference of
an ellipse?
3. Is there also a foci of a parabola?(6 votes)
1. The area of an ellipse is easy; if the major and minor radii are a and b, the area is just πab. Notice that in the special case of a circle, where a=b=r, this just becomes πr^2, as you would expect.
The circumference of an ellipse, interestingly, is much harder to calculate. There is not a simple formula for it, and the only way to calculate it is by approximating it with calculus.
2. π is a part of the expression for the area of an ellipse, yes. The circumference of an ellipse, again, can only be approximated. If you can approximate an integral, you don't need any mention of π.
3. Yes, a parabola has one focus. You can think of a parabola as an ellipse where one of the two foci has stayed fixed while the other focus moved out to infinity.(20 votes)
Why do we need ellipses in the real world? Only to describe shapes or for something greater?(5 votes)
Conic sections in general are really common shapes in calculus. And knowing the general form of ellipses, circles, parabolas etc. helps a lot when you need to identify them while integrating over surfaces.
Plus, they're cool shapes! Definitely better than a boring, straight line :)(8 votes)
- No questions but why do we learn this(3 votes)
- Conic sections are incredibly useful when it comes to some calculus related concepts, as many of your shapes at that point will be stuff like ellipses, hyperbolas etc.(1 vote)
Why no section on parabolas?(1 vote)- i can see conic sections like circles, ellipses, and parabolas have real life applications, but never hyperbolas. in what scenarios (except from the example) can i see applications of hyperbolas it in real life?(1 vote)
