Main content
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 points on either side of the center, which are used to generate the hyperbola by defining a fixed distance from each focus to any point on the hyperbola.
- 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?
No posts yet.