Can anybody explain if the reciprocal cone of the cone touching the three axes will be the cone touching the three coordinate planes. If yes, why?
$3$-D diagram would really help if possible. Thanks.
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It turned out that the reciprocal cone is just what you get taking the inverse of the symmetric matrix that defined the first cone. I gave a big answer at condition for cones to be reciprocal
I'm not going to post any diagrams.
The "cone touching the three axes" is $$ yz + zx + xy = 0. $$ The matrix can be the Hessian matrix of second partial derivatives of the quadratic form. The basic rule is that the Hessian matrix have determinant not equal to zero.
