Degenerate conics are obtained when the slicing plane cuts the double right circular cone at the intersecting tips of the two cones (vertex). This gives a pair of straight lines intersecting at the vertex (depending upon the eccentricity). Dandelin spheres are useful in finding the focus, directrix, etc.

Focus of a straight line lies at infinity. I don't know what happens to the directrix of the conic when it approaches becoming degenerate. I think the Dandelin sphere gets smaller and smaller when the slicing plane moves towards the vertex of the two cones, but I am unable to explain this in relation to the focus and directrix.

So, what will happen to Dandelin Spheres, and their applications (finding focus, directrix, etc.,) in case of degenerate conics? Or is that not defined at all for such cases.


As thhe Dandelin spheres shrink to the vertex, the foci are driven to do the same and the directrices are also forced to pass through the vertex. But how are the directtuces oriented?

With a regular conic intersection, when you draw the directrices you find theyvare skew perpendicular to the axis, acresult of the spheres being centered on this axis.

Such skew perpendicular lines, when degenerated into the gertex, becone a perpenducular line to the axis that now lies in the plane (which is also passing through the vertex). This lihe us unique for a given cone and plane, except fir tge xase of a circular criss-section where the directrices would never be defined anyway. So the degenerated directrix is the line in the cutting plane that passes through the vertex and is perpendicuar to the axis.

We may also consider what we usually consider a nondegenerate case, the parabola. To make a parabola you orient the planr paralkel to a linear element of the cone. This forces the intersection of that eldment with the plane to infinity. The Dandelin sphere passing near to that intersection, and the associated focus and directrix, are also driven to infinity. That's how a parabola shows only one remaining focus/directrix pair as points and lines in "visible" locations.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.