Shadow of an Egg
1 6 x 2 + 2 5 y 2 + 9 z 2 = 1
There is a point source of light above the ellipsoid. The section of the shadow cast by the ellipsoid by the plane z = 0 is a circle. If the locus of the point source is a central conic. Find the product of the semi conjugate and the semi transverse axis.
The answer is 9.
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1 solution
It should be (a^2/16 + b^2/25 + c^2/9 -1). Sorry for the mistake. Please do not mind the fact that I dont want to write solution again.
I was looking for more information on the enveloping cone, and I found this post: https://math.stackexchange.com/questions/2880013/the-section-of-the-enveloping-cone-of-the-ellipsoid-whose-vertex-is-p-by-the .
It gives the equation of the enveloping cone as ( a 2 x 2 + b 2 y 2 + c 2 z 2 − 1 ) ( a 2 x 1 2 + b 2 y 1 2 + c 2 z 1 2 − 1 ) = ( a 2 x x 1 + b 2 y y 1 + c 2 z z 1 − 1 ) 2 , which is different from your equation.
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It is basically the same as SS1 = T^2(like we do in 2D geometry when we find out the equation of a pair of tangents form a given point to a conic) format. I wrote it in a differnet way. But it equates out the same. (see my commet that in the equation I forgot to write the -1 in the factor (a^2/16 + b^2/25 + c^2/9 -1) . Mr. Hosam Hajjir is the only one to solve this problem other than me till now and he also pointed out an initial mistake in finding the locus which I updated later. You can ask him if I was unable to explain it