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

# Right circular cylinder with guiding circle

Mathematics · Chapter 3: Three Dimensional Solid Geometry

- Asked in: 2075 Baisakh, 5 marks; 2067 Mangsir, 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-guiding-circle
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-3

## Question

Find the equation of the right circular cylinder whose guiding curve is the circle x²+y²+z²−x−y−z = 0, x+y+z = 1.

## Short answer

Axis direction (1,1,1), through circle centre (1/3,1/3,1/3), radius²=2/3. For any point, distance squared to axis: Σ(x−1/3)² − (x+y+z−1)²/3 = 2/3. Simplifies to x²+y²+z²−xy−yz−zx=1.

## Answer

**Given.** Guiding circle: sphere S: x² + y² + z² − x − y − z = 0 and plane x + y + z = 1.

A right circular cylinder has its axis perpendicular to the plane of its guiding circle, passing through the centre of the circle, and its radius equals the circle's radius.

#### Axis
The plane's normal is (1, 1, 1), so the axis has direction ratios (1, 1, 1).

Sphere: centre (1/2, 1/2, 1/2), radius R = √(3/4) = √3/2.

Distance from the sphere's centre to the plane: p = |3/2 − 1|/√3 = 1/(2√3), so p² = 1/12.

Circle radius: r² = R² − p² = 3/4 − 1/12 = 2/3.

The circle's centre is the foot of the perpendicular from (1/2, 1/2, 1/2): points (1/2 + t, 1/2 + t, 1/2 + t) with 3/2 + 3t = 1, t = −1/6. Centre = (1/3, 1/3, 1/3).

So the axis is (x − 1/3)/1 = (y − 1/3)/1 = (z − 1/3)/1.

#### Equation of the cylinder
A point P(x, y, z) lies on the cylinder if its distance from the axis is r:

|PA|² − (PA·n)²/|n|² = r², with A = (1/3, 1/3, 1/3), n = (1, 1, 1).

```text
Σ(x − 1/3)² − (x + y + z − 1)²/3 = 2/3
x² + y² + z² − (2/3)s + 1/3 − (s² − 2s + 1)/3 = 2/3      (s = x + y + z)
3(x² + y² + z²) − 2s + 1 − s² + 2s − 1 = 2
3(x² + y² + z²) − (x + y + z)² = 2
2(x² + y² + z²) − 2(xy + yz + zx) = 2
```

**Answer: x² + y² + z² − xy − yz − zx = 1.**

## 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?
   A. (1, 1, 0)
   B. (1, 1, 1), the normal of the plane of the circle
   C. (1, -1, 1)
   D. The direction through the sphere's centre only
   Answer: B. A right circular cylinder's axis is perpendicular to the plane of the guiding circle. (Key point: Axis direction)

2. What is r^2, the square of the circle's radius?
   A. 2/3
   B. 3/4
   C. 1/12
   D. 1/3
   Answer: A. r^2 = R^2 - p^2 = 3/4 - 1/12 = 2/3. (Key point: Radius of circle)

3. What is the centre of the guiding circle?
   A. (1/2, 1/2, 1/2)
   B. (1, 1, 1)
   C. (1/3, 1/3, 1/3)
   D. (0, 0, 0)
   Answer: C. The foot of the perpendicular from (1/2, 1/2, 1/2) to x + y + z = 1 has t = -1/6. (Key point: Centre of circle)

4. What is the equation of the cylinder?
   A. x^2 + y^2 + z^2 - xy - yz - zx = 1
   B. x^2 + y^2 + z^2 + xy + yz + zx = 1
   C. x^2 + y^2 + z^2 - xy - yz - zx = 2/3
   D. x^2 + y^2 + z^2 - xy - yz - zx = 2
   Answer: A. From 3(x^2+y^2+z^2) - (x+y+z)^2 = 2 we get 2(x^2+y^2+z^2) - 2(xy+yz+zx) = 2. (Key point: Final equation)

## Close variants set in other exams

- [Right circular cylinder with guiding circle x²+y²+z²=9](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aright-circular-cylinder-guiding-circle-2) — Mathematics, chapter 3, asked 3×

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