Question

Find the equation of a cone reciprocal to the cone ax2+by2+cz2=0

Answer 1

Reciprocal cone formula:

We consider the equation of a cone in homogeneous of second degree, which is of the form

ax² + by² + cz² + 2fyz + 2gzx + 2hxy = 0

Another cone

Ax² + By² + Cz² + 2Fyz + 2Gzx + 2Hxy = 0

can be called the reciprocal cone to the first cone, when

A = bc - f² ,

B = ca - g² ,

C = ab - h² ,

F = gh - af ,

G = hf - bg and

H = fg - ch.

Solution:

The given cone is

ax² + by² + cz² = 0

Then the reciprocal cone must be determined using a, b, c in A, B, C, F, G, H (since f, g, h = 0 here).

A = bc - f² = bc ,

B = ca - g² = ca ,

C = ab - h² = ab and

F = G = H = 0

Hence, the reciprocal cone is

bcx² + cay² + abz² = 0

or, x²/a + y²/b + z²/c = 0 [ abc ≠ 0 ]