Right circular cone with vertex (1,1,1), semi-vertical 30°

Given. Vertex V(1, 1, 1), axis (x − 1)/1 = (y − 1)/2 = (z − 1)/3, semi-vertical angle α = 30°.

Condition

For a point P(x, y, z) on the cone, the line VP makes angle α with the axis. Write X = x − 1, Y = y − 1, Z = z − 1, so VP = (X, Y, Z), and the axis direction is (1, 2, 3) with length √14.

cos α = (X + 2Y + 3Z) / ( √14 · √(X² + Y² + Z²) )
(X + 2Y + 3Z)² = 14 cos²α (X² + Y² + Z²)

With α = 30°, cos²α = 3/4, so 14 cos²α = 21/2:

2(X + 2Y + 3Z)² = 21 (X² + Y² + Z²)

Expand

(X + 2Y + 3Z)² = X² + 4Y² + 9Z² + 4XY + 12YZ + 6ZX.

2X² + 8Y² + 18Z² + 8XY + 24YZ + 12ZX = 21X² + 21Y² + 21Z²
19X² + 13Y² + 3Z² − 8XY − 24YZ − 12ZX = 0

In terms of x, y, z

Replace X, Y, Z by x − 1, y − 1, z − 1 and expand:

19x² + 13y² + 3z² − 8xy − 24yz − 12zx − 18x + 6y + 30z − 9 = 0

Check: the vertex (1, 1, 1) gives 19 + 13 + 3 − 8 − 24 − 12 − 18 + 6 + 30 − 9 = 0 ✓.