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The following image is from the Wikipedia Article on Conic Section:

enter image description here

Where are the focus, vertex and the directrix in the above diagram? I know they must lie on the plane which cuts the right circular cone, but I am unable to determine their position.

For the circle in the lower cone in $(2)$, I concluded that the centre is the point where the axis of the cones intersect the slicing plane due to symmetry reasons. But things get tricky when I move on to other conics.

Kindly explain in a simple way that could be understood by a High School student. Thank you in advance.

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    $\begingroup$ Take a look at Dandelin spheres. $\endgroup$ – Blue Nov 7 '19 at 7:28
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    $\begingroup$ @Blue, Thanks! That told where the focus is. Could you please reveal where the directrix and the vertex are? I can guess the axis of the cone is not same as that of the conic. But, once these two are specified, I'll be able to determine the axis on my own :) $\endgroup$ – Guru Vishnu Nov 7 '19 at 7:33
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    $\begingroup$ Keep reading: the first paragraph of the “Proof of the focus-directrix property” section of that Wikipedia article tells you how to find the directrices using Dandelin spheres. $\endgroup$ – amd Nov 7 '19 at 7:41
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    $\begingroup$ They touch the cone at more than one point. $\endgroup$ – amd Nov 7 '19 at 7:49
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    $\begingroup$ The white curves labeled $k_1$ and $k_2$ in that very figure are the intersections of the two spheres with the cone, which are circles. Try a physical experiment for yourself: make a cone from a piece of paper and drop a small spherical object into it. It doesn’t fall or roll all the way to the vertex, so clearly it must contact the cone at more than one point. $\endgroup$ – amd Nov 7 '19 at 7:58
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I wish to express my sincere thanks to @Blue and @amd for clearing my doubt in the comments.


The location of the focus, vertex, and directrix of a conic can be easily determined by knowing some basics about Dandelin Spheres. Wikipedia gives the following explanation:

In geometry, the Dandelin spheres are one or two spheres that are tangent both to a plane and to a cone that intersects the plane. The intersection of the cone and the plane is a conic section, and the point at which either sphere touches the plane is a focus of the conic section, so the Dandelin spheres are also sometimes called focal spheres.

enter image description here

In the above diagram, the yellow plane cuts the blue cone forming an ellipse. Next, imagine inserting two spheres of maximum volume in the upper and lower parts of the cone demarcated by the slicing plane, such that they just touch the surfaces (curved surface of the cone and slicing plane) but don't peep out. These two are called as Dandelin spheres. The points at which these spheres touch the yellow plane (slicing plane), $F_1$ and $F_2$ are the foci of the ellipse. So now, we have located the foci of the ellipse.

The same Dandelin Spheres are helpful in determining the directrix and the vertex. The two spheres in the above diagram, touch the curved surface of the cone in a circle (represented by white circles $k_1$ and $k_2$. Let us consider, two planes passing through these two circles separately. These planes meet the yellow plane (slicing plane) in straight lines (unless and until the yellow plane cuts out a circle in the cone, when it's parallel to the base of cone). The two lines formed by the intersection of the three planes (slicing plane, and the two planes through the two circles) are parallel to each other. These lines are the directrices of the ellipse.

enter image description here

The above [.gif] explains the same visually. In this, the light blue plane is the slicing plane, orange sphere is one of the Dandelin spheres, transparent planes are those which pass through the circular region formed by the intersection of the spheres and the curved surface of the cone. Here the slicing plane, forms an ellipse which is shown by blue. The two parallel blue lines are the directrices.

Even though I've explained using ellipses, the same concept can be extended to other ellipses. For example, parabola since it has only one focus, has only one Dandelin sphere. Hyperbola has two foci in opposite nappes, so it has two such spheres in the two nappes. Using this we can determine the directrix and the focus of the conic.

Now, coming to the last part of the answer, finding the vertex. This is simple once we've found the directrix and the focus. Just draw a line perpendicular to the directrix passing though the focus. This line is the axis of the conic (and not that of the cone!). The point where axis meets the curve is the vertex.

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