---
title: "Right circular cylinder of radius 2 (axis 1,1,2)"
subject: "Mathematics"
chapter: 3
key: "semester_2/mathematics:right-circular-cylinder-radius-2-2"
source: https://www.bctengineer.com/q/semester_2/mathematics/right-circular-cylinder-radius-2-2
site: BCT Engineering
---

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

Mathematics · Chapter 3: Three Dimensional Solid Geometry

- Asked in: 2073 Bhadra, 5 marks; 2070 Bhadra, 5 marks (asked 2 times)
- Video: [SOLID GEOMETRY | Cylinder | Equation of Right Circuar Cylinder | Lecture 06 | PRADEEP GIRI SIR](https://www.youtube.com/watch?v=lUepc_KwOBs) — Pradeep Giri Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/right-circular-cylinder-radius-2-2
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-3

## Question

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.

## Short answer

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

## Answer

**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.

#### Condition
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

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

#### Expand
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.

```text
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.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What are d and |d|^2 for the axis (x-1)/1 = (y-2)/1 = (z-3)/2?
   A. d = (1, 1, 2), |d|^2 = 6
   B. d = (1, 1, 2), |d|^2 = 4
   C. d = (1, 2, 3), |d|^2 = 14
   D. d = (1, 1, 2), |d|^2 = 36
   Answer: A. |d|^2 = 1 + 1 + 4 = 6. (Key point: Axis data)

2. After multiplying by |d|^2, which is the condition?
   A. 6 Sum (x - x0)^2 - M^2 = 4
   B. 6 Sum (x - x0)^2 - M^2 = 24
   C. Sum (x - x0)^2 - M^2 = 24
   D. 6 Sum (x - x0)^2 + M^2 = 24
   Answer: B. Radius^2 x |d|^2 = 4 x 6 = 24. (Key point: Cleared equation)

3. What is the coefficient of z^2 in the final equation?
   A. 6
   B. 5
   C. 2
   D. 4
   Answer: C. 6 from the first term minus 4 from M^2 gives 2. (Key point: Coefficient of z^2)

4. What is the constant term?
   A. 84
   B. -81
   C. 21
   D. -21
   Answer: D. 84 - 81 - 24 = -21. (Key point: Final equation)

## Close variants set in other exams

- [Right circular cylinder of radius 2 (axis 2,1,2)](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aright-circular-cylinder-radius-2) — Mathematics, chapter 3, asked 7×
- [Right circular cylinder with axis (x−α)/l=(y−β)/m=(z−γ)/n](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aright-circular-cylinder-axis-x) — Mathematics, chapter 3, asked 7×
- [Right circular cylinder of radius 4](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aright-circular-cylinder-radius-4) — Mathematics, chapter 3, asked 7×

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