Right circular cylinder of radius 2 (axis 1,1,2)

Short answer

Using distance from a point to the axis, the equation is 5x² + 5y² + 2z² − 2xy − 4yz − 4zx + 6x − 6y − 21 = 0.

Find the equation of the right circular cylinder of radius 2 whose axis is the line (x−1)/1 = (y−2)/1 = (z−3)/2.

Given. Axis: (x − 1)/1 = (y − 2)/1 = (z − 3)/2, radius 2.

The axis passes through A(1, 2, 3) with direction d = (1, 1, 2), |d|² = 1 + 1 + 4 = 6.

A point P(x, y, z) lies on the cylinder if its distance from the axis is 2. With AP = (x − 1, y − 2, z − 3):

|AP|² − (AP·d)²/|d|² = 4

6[(x − 1)² + (y − 2)² + (z − 3)²] − [(x − 1) + (y − 2) + 2(z − 3)]² = 24

Let M = x + y + 2z − 9. Then M² = x² + y² + 4z² + 2xy + 4xz + 4yz − 18x − 18y − 36z + 81.

6[(x−1)² + (y−2)² + (z−3)²] = 6x² + 6y² + 6z² − 12x − 24y − 36z + 84.

6Σ − M² − 24 = 0
5x² + 5y² + 2z² − 2xy − 4xz − 4yz + 6x − 6y + (84 − 81 − 24) = 0

Answer: 5x² + 5y² + 2z² − 2xy − 4yz − 4zx + 6x − 6y − 21 = 0.

✓ Answer checked by Claude Opus 5.5 (high effort)

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