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

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

Mathematics · Chapter 3: Three Dimensional Solid Geometry

- Asked in: 2074 Bhadra, 5 marks; 2073 Magh, 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
- 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)/2 = (y−2)/1 = (z−3)/2.

## Short answer

Axis through (1,2,3) with direction (2,1,2). Distance condition: |AP|² − (AP·d)²/9 = 4. With AP = (x−1,y−2,z−3), expanding yields 5x²+8y²+5z²−4xy−4yz−8zx+22x−16y−14z−10=0.

## Answer

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

The axis passes through A(1, 2, 3) with direction d = (2, 1, 2), |d|² = 9.

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

distance² = |AP|² − (AP·d)²/|d|² = 4

```text
9[(x − 1)² + (y − 2)² + (z − 3)²] − [2(x − 1) + (y − 2) + 2(z − 3)]² = 36
```

#### Expand
Let L = 2x + y + 2z − 10, so L² = 4x² + y² + 4z² + 4xy + 8xz + 4yz − 40x − 20y − 40z + 100.

9[(x−1)² + (y−2)² + (z−3)²] = 9x² + 9y² + 9z² − 18x − 36y − 54z + 126.

Subtract and equate to 36:

```text
5x² + 8y² + 5z² − 4xy − 8xz − 4yz + 22x − 16y − 14z + 126 − 100 − 36 = 0
```

**Answer: 5x² + 8y² + 5z² − 4xy − 4yz − 8zx + 22x − 16y − 14z − 10 = 0.**

## Check yourself

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

1. What are the point and |d|^2 for the axis (x-1)/2 = (y-2)/1 = (z-3)/2?
   A. Point (1, 2, 3), |d|^2 = 9
   B. Point (1, 2, 3), |d|^2 = 3
   C. Point (2, 1, 2), |d|^2 = 9
   D. Point (-1, -2, -3), |d|^2 = 5
   Answer: A. d = (2, 1, 2), so |d|^2 = 4 + 1 + 4 = 9. (Key point: Axis data)

2. Which condition describes points on the cylinder?
   A. |AP|^2 + (AP . d)^2/|d|^2 = 4
   B. |AP|^2 - (AP . d)^2/|d|^2 = 4
   C. |AP|^2 = 4
   D. (AP . d)^2/|d|^2 = 4
   Answer: B. The squared perpendicular distance from the axis is |AP|^2 minus the squared projection on d. (Key point: Distance condition)

3. After multiplying by 9, what is the right side?
   A. 4
   B. 9
   C. 36
   D. 18
   Answer: C. 4 x 9 = 36. (Key point: Cleared form)

4. What is the coefficient of x^2 in the final equation?
   A. 5
   B. 9
   C. 4
   D. 13
   Answer: A. 9 from the first term minus 4 from L^2 gives 5. (Key point: Final equation)

## Close variants set in other exams

- [Right circular cylinder of radius 2 (axis 1,1,2)](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aright-circular-cylinder-radius-2-2) — 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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