Right circular cone, vertical angle 30°

Given. Vertex O(0, 0, 0), axis x/1 = y/2 = z/3, vertical angle 30°.

Assumption. The vertical angle is the full angle at the vertex, so the semi-vertical angle is α = 15°.

Method

For a point P(x, y, z) on the cone, the line OP makes angle α with the axis, whose direction is (1, 2, 3):

cos α = (x + 2y + 3z) / ( √14 · √(x² + y² + z²) )
(x + 2y + 3z)² = 14 cos²α (x² + y² + z²)

Value of cos² 15°

cos² 15° = (1 + cos 30°)/2 = (1 + √3/2)/2 = (2 + √3)/4.

(x + 2y + 3z)² = 14 · (2 + √3)/4 · (x² + y² + z²)
2 (x + 2y + 3z)² = 7 (2 + √3)(x² + y² + z²)

Expanded form

Expanding (x + 2y + 3z)² = x² + 4y² + 9z² + 4xy + 12yz + 6zx and collecting terms:

(12 + 7√3)x² + (6 + 7√3)y² + (7√3 − 4)z² − 8xy − 24yz − 12zx = 0

(If the 30° were the semi-vertical angle, replace cos²15° by cos²30° = 3/4. This gives 2(x + 2y + 3z)² = 21(x² + y² + z²), i.e. 19x² + 13y² + 3z² − 8xy − 24yz − 12zx = 0.)