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### : Analytical geometry: find the point of touch of ellipsoid and cone

let the point p (xp, yp, zp), and the ellipsoid, which are known to the semi-axis and the center, its equation: x ^ 2 /a ^ 2 + y ^ 2 /a ^ 2 + z ^ 2 /b ^ 2= 1. Point P lies outside the ellipsoid in some distance. A set of tangents to the ellipsoid carried out from P forms a conical surface. It is necessary to find a general view of the point of touching this cone and ellipsoid (it is clear that these points will belong to the ellipse, that is, the task is essentially reduced to finding the cone and ellipsoid cone ellipse equation). Is it possible to simplify the task here by going to spherical coordinates? If so, how to act next? If not, then the question is similar.

The property "to be tangent" does not change in compression-stretching axes. Therefore, you can go to the coordinate system in which the ellipse is a sphere, find the equation of tangent to this area, and stretch the coordinates back.

Pak Uula2021-06-13 04:11:11

Well, but how to be next? It is clear that the task will be simplified, but I still have no idea how to move on.

Fire13nyu2021-06-13 04:11:11