Right circular cylinder of radius 4

Given. Axis x = 2y = −z, radius 4.

Axis

Put x = 2y = −z = 2k. Then x = 2k, y = k, z = −2k. So the axis passes through the origin with direction ratios d = (2, 1, −2), and |d|² = 4 + 1 + 4 = 9.

Condition

A point P(x, y, z) lies on the cylinder if its distance from the axis is 4. Since the axis passes through O, OP = (x, y, z) and

distance² = |OP|² − (OP·d)²/|d|² = 16

(x² + y² + z²) − (2x + y − 2z)²/9 = 16
9(x² + y² + z²) − (2x + y − 2z)² = 144

Expand

(2x + y − 2z)² = 4x² + y² + 4z² + 4xy − 8xz − 4yz.

9x² + 9y² + 9z² − 4x² − y² − 4z² − 4xy + 8xz + 4yz = 144

Answer: 5x² + 8y² + 5z² − 4xy + 8xz + 4yz = 144.