---
title: "Right circular cylinder of radius 4"
subject: "Mathematics"
chapter: 3
key: "semester_2/mathematics:right-circular-cylinder-radius-4"
source: https://www.bctengineer.com/q/semester_2/mathematics/right-circular-cylinder-radius-4
site: BCT Engineering
---

# Right circular cylinder of radius 4

Mathematics · Chapter 3: Three Dimensional Solid Geometry

- Asked in: 2071 Magh, 5 marks; 2069 Poush, 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-4
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-3

## Question

Obtain the equation of right circular cylinder of radius 4 and axis the line x = 2y = −z.

## Short answer

The cylinder equation is 5x² + 8y² + 5z² − 4xy + 8xz + 4yz = 144.

## Answer

**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

```text
(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.

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

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

## Check yourself

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

1. What is the direction of the axis x = 2y = -z?
   A. (2, 1, -2) through the origin
   B. (1, 2, -1) through the origin
   C. (2, 1, 2) through the origin
   D. (1, 1, 1) through (1, 0, 0)
   Answer: A. Setting x = 2y = -z = 2k gives (2k, k, -2k). (Key point: Axis direction)

2. What equation holds for a point on the cylinder?
   A. 9(x^2 + y^2 + z^2) - (2x + y - 2z)^2 = 16
   B. 9(x^2 + y^2 + z^2) - (2x + y - 2z)^2 = 144
   C. (x^2 + y^2 + z^2) - (2x + y - 2z)^2 = 144
   D. 9(x^2 + y^2 + z^2) + (2x + y - 2z)^2 = 144
   Answer: B. |OP|^2 - (OP . d)^2/9 = 16, and multiplying by 9 gives 144. (Key point: Condition)

3. What is the coefficient of xz in the final equation?
   A. -8
   B. +4
   C. +8
   D. 0
   Answer: C. The term -(2x + y - 2z)^2 contains -(-8xz) = +8xz. (Key point: Coefficients)

## 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 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×
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