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Course: High school geometry>Unit 7

Lesson 1: Graphs of circles intro

Intro to conic sections

Sal introduces the four conic sections and shows how they are derived by intersecting planes with cones in certain ways. Created by Sal Khan.

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• Will two same parabolas, placed side to side ( like this <>) , make a perfect ellipse ?
• Nope. They will have a sharp angle in the joint. If you see, for them to fit, a parabola must be vertical at some point, which is actually not true, it gets always steeper but don't get vertival until reaching infinity.

However, If you considered possible to reach infinity, then placing them infinitely separated could make them actually fit, but I doubt that shape would form an ellipse. (By the way, all shapes are either ellipses or aren't ellipses at all. There is no such a thing as a perfect ellipse : )
• So these conic sections are 2D shapes that intersect a 3D cone? Are there 3 dimensional shapes that intersect 4 dimensional shapes?
• Camerxn, you're right that time is a temporal dimension, not a spatial one, although ANY method of distinguishment (color, musical note, shape, etc.) could be a dimension, however '4D' is considered to be 4 spatial dimensions (x axis, y axis, z axis, w axis).

The more dimensions (particularly spatial ones) the more fun you can have. You can define the foci for an ellipse by using a cone, two spheres and zero math. In four dimensions, I'm willing to bet that there are ways of defining 3-dimensional shapes in a similar manner (I can't think of any off the top of my head). For the cone, make a sphere just big enough to touch the desired ellipse at one point inside the cone, and the other sphere just small enogh to touch the same ellipse in a second point, nestled on top of the cone (think of an Ice cream cone), those two points are the foci.

Three dimensional knots wouldn't but four dimensional knots would (think of a bow knot with 3 bows instead of two, something like that I think).
• Can't you also get a triangle by going perpendicular to the base of the cone and going through the tip?
• Good catch! It's not quite a triangle, because the theoretical cones in this example have infinite height, but it is two intersecting straight lines. Other "weird" examples are a single point, a single straight line, and two parallel straight lines. (See if you can figure out how to get those!) All of these examples are called degenerate conic sections, and we will never speak of them again. ^_^
• From 2009, what a classic.
• Dose the Hyperbola ever touch the asmpotote?
• No, hyperbolas never reach the asymptotes, which is why they are called asymptotes. As the hyperbola gets further and further away from the center, the hyperbola approaches the asymptotes, but is unable to touch it.
• So I understand how the circle, ellipse, parabola, and the hyperbola are related, but I still don't understand what the meaning of conic sections is.
• To 'section' something is to cut it... like in biology class when you 'dissect' a frog.

Imagine a cone... an ice cream cone (with no ice cream) or one of those orange cones they put around utility vehicles in the street.

If you hold such a cone so its central axis is vertical, then cut it with a horizontal plane, the cut edge will be a circle. If you tilt the cutting plane a bit, but not so much that it is parallel to the outer edge of the cone and section the cone the cut edge will be an ellipse. If you section the cone with a plane that is parallel to the outer surface of the cone the cut edge will be a parabola and if you tilt the cutting plane past that point and on to vertical you will get a hyperbola.

So the 'conic sections' are literally the shapes you get when you section a cone.

See http://en.wikipedia.org/wiki/Conic_section for a lot more detail.
• At , Sal draws a "cone", however it appears to be two cones on top of each other (tip to tip). Why is this?
• The correct term for the solid is "double-napped cone". Essentially two congruent cones with the same axis and a common vertex. Sal just didn't use the formal name for the solid.
• Can I differentiate or integrate conic section (circle, parabola, ellipse, hyperbola) equations?
• Yes, absolutely, if you want to learn partial differentiation early! Since there are two variables in these equations, you can take two separate derivatives. When you take the partial derivative of the function for the X variable, consider Y a constant (and vice versa). For example, given `x^2 + y^2` then the partial derivative with respect to X would be just 2x + 0. The result is the change in the X direction when Y is not changing at all. This concept is built upon in Multivariable Calculus and is really fun :)
• I have a question about how Sal got the parabola form the cone: If both the plane and the cone go on infinitely, shouldn't the plane intercept the cone again and therefore close the parabola (and form an ellipse instead)? From my understanding, The only way the parabola could remain open is if the plane was parallel to the cone but translated just so it only meets half of the cone.
• > The only way the parabola could remain open is if the plane was parallel to the cone but translated just so it only meets half of the cone.

That's exactly what he does. At :

> I'm drawing it in such a way that it only intersects this bottom cone as the surface of the plane is parallel to the side of this top cone.

If the plane were angled less than parallel, it would intersect the bottom cone twice and become an ellipse. If the plane were angled more than parallel, it would also intersect the top cone and become a hyperbola.