# Conic section

And that's frankly where the word conic comes from. Well, I will because you're probably familiar with it. Eccentricity of conic sectionsThe eccentricity of a conic section completely characterizes its shape. Let me see if I can do it justice. Let me see if I can draw that.

If you keep kind of tilting-- if you start with a circle, tilt a little bit, you get an ellipse. This is the front line that's closer to you and then they would have another line back here. But a parabola-- let me draw a line here to separate things. If this plane is directly perpendicular to the axis of these and this is where the plane goes behind it. So this is a cone. The line joining the foci is called the principal axis and the points of intersection of the conic with the principal axis are called the vertices of the conic.

So let me see if I can draw a three-dimensional cone. So if my plane looks like this-- I know it's very hard to read now-- and you wanted the intersection of this plane, this green plane and the cone-- I should probably redraw it all, but hopefully you're not getting overwhelmingly confused-- the intersection would look like this. So to put things simply because they're the intersection of a plane and a cone. A circle has an eccentricity of zero, so the eccentricity shows us how "un-circular" the curve is. But a hyperbola would look something like this.

Now, if we take the plane and we tilt it down a little bit, so if instead of that we have a situation like this. A property that the conic sections share is often presented as the following definition.

So the one thing that I'm sure you're asking is why are they called conic sections. These are called degenerate conics and some authors do not consider them to be conics at all. This would be intersection of the plane and the bottom cone. The midpoint of this line segment is called the center of the conic. Circle, we all know what that is. There are three types of conics, the ellipseparabolaand hyperbola.

In contrast, ellipses and hyperbolas vary greatly in shape. They're the circle, the ellipse, the parabola, and the hyperbola. There are three types of conics, the ellipseparabolaand hyperbola.

By a suitable choice of coordinate axes, the equation for any conic can be reduced to one of three simple r forms: The circle is obtained when the cutting plane is parallel to the plane of the generating circle of the cone — for a right cone, see diagram, this means that the cutting plane is perpendicular to the symmetry axis of the cone.

And I'll switch colors. Eighteenth-century architects created a fad for whispering galleries —such as in the U. So if this is my new plane-- whoops. See you in the next video.

So now my plane-- let me see if I can do this. Planes that pass through the vertex of the cone will intersect the cone in a point, a line or a pair of intersecting lines.

So let's take different intersections of a plane with this cone and see if we can at least generate the different shapes that we talked about just now. If the cutting plane is parallel to exactly one generating line of the cone, then the conic is unbounded and is called a parabola.

Archimedes is said to have used this property to set enemy ships on fire. There, that should do it. So an ellipse could be like that. Definition The black boundaries of the colored regions are conic sections. The plane would look something like this.

In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse.

This lesson, and the conic-specific lessons to which this page links, will instead concentrate on: finding curves, given points and other details; finding points and other details, given curves; and setting up and solving conics equations to solve typical word problems.

The pedal curve of a conic section with pedal point at a focus is either a circle or a line. In particular the ellipse pedal curve and hyperbola pedal curve are both circles, while the parabola pedal curve is a line (Hilbert and Cohn-Vossenpp.

). Five points in a plane determine a conic. Learn about the four conic sections and their equations: Circle, Ellipse, Parabola, and Hyperbola. Site Navigation Our mission is to provide a free, world-class education to anyone, anywhere. Algebra conic sections lessons with lots of worked examples and practice problems.

Very easy to understand! This lesson, and the conic-specific lessons to which this page links, will instead concentrate on: finding curves, given points and other details; finding points and other details, given curves; and setting up and solving conics equations to .

Conic section
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Conic sections | Precalculus | Math | Khan Academy