---
title: "Right circular cylinder with axis (x−α)/l=(y−β)/m=(z−γ)/n"
subject: "Mathematics"
chapter: 3
key: "semester_2/mathematics:right-circular-cylinder-axis-x"
source: https://www.bctengineer.com/q/semester_2/mathematics/right-circular-cylinder-axis-x
site: BCT Engineering
---

# Right circular cylinder with axis (x−α)/l=(y−β)/m=(z−γ)/n

Mathematics · Chapter 3: Three Dimensional Solid Geometry

- Asked in: 2069 Bhadra, 5 marks
- Video: [3D Geometry - CYLINDER | The Right Circular CYLINDER - Concept & Example | By Gajendra Purohit](https://www.youtube.com/watch?v=tZskoY3waAQ) — Dr.Gajendra Purohit
- 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-axis-x
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-3

## Question

Find the equation of right circular cylinder whose axis is the line (x−α)/l = (y−β)/m = (z−γ)/n and whose radius is 'r'.

## Short answer

The set of points whose perpendicular distance from the axis equals r is a right circular cylinder. One form is (x-α)^2+(y-β)^2+(z-γ)^2 - [l(x-α)+m(y-β)+n(z-γ)]^2/(l^2+m^2+n^2) = r^2.

## Answer

A right circular cylinder is the surface traced by a line parallel to its axis at a constant distance *r* from the axis. So a point P(x, y, z) lies on it if its perpendicular distance from the axis equals *r*.

Let the axis pass through A(α, β, γ) with direction ratios (l, m, n).

#### Distance of P from the axis
Let AP = (x − α, y − β, z − γ). The length of the projection of AP on the axis is

AN = [l(x − α) + m(y − β) + n(z − γ)] / √(l² + m² + n²).

By Pythagoras in the right triangle APN, the perpendicular distance PN satisfies

```text
PN² = AP² − AN²
```

#### Condition PN = r
```text
(x−α)² + (y−β)² + (z−γ)²
   − [l(x−α) + m(y−β) + n(z−γ)]² / (l²+m²+n²) = r²
```

#### Equivalent form
Since |AP|²|d|² = (AP·d)² + |AP × d|² (Lagrange's identity), the same equation can be written as

```text
[n(y−β) − m(z−γ)]² + [l(z−γ) − n(x−α)]² + [m(x−α) − l(y−β)]²
      = r²(l² + m² + n²)
```
(the left side is |AP × axis direction|², which is (distance × |direction|)²).

**Answer: the required cylinder is the surface given by either form above.**

Example: axis = z-axis (α = β = γ = 0, l = m = 0, n = 1) gives x² + y² = r², as expected.

## Check yourself

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

1. A point P lies on the right circular cylinder when:
   A. Its distance from A equals r
   B. Its perpendicular distance from the axis equals r
   C. Its projection on the axis equals r
   D. AP is parallel to the axis
   Answer: B. All points of a right circular cylinder are at constant distance r from the axis. (Key point: Defining property)

2. In right triangle APN with AN the projection of AP on the axis, PN^2 equals:
   A. AP^2 + AN^2
   B. AN^2 - AP^2
   C. AP^2 - AN^2
   D. AP * AN
   Answer: C. PN is the perpendicular leg, AP the hypotenuse. (Key point: Pythagoras step)

3. What is AN?
   A. [l(x-α) + m(y-β) + n(z-γ)] / sqrt(l^2+m^2+n^2)
   B. [l(x-α) + m(y-β) + n(z-γ)]
   C. sqrt(l^2+m^2+n^2)
   D. [l(x-α) + m(y-β) + n(z-γ)] / (l^2+m^2+n^2)
   Answer: A. AN = AP . d / |d|. (Key point: Projection length)

4. For the z-axis (l = m = 0, n = 1, through the origin) the equation reduces to:
   A. x^2 + y^2 + z^2 = r^2
   B. x^2 + y^2 = r^2
   C. z^2 = r^2
   D. x + y = r
   Answer: B. The distance from the z-axis is sqrt(x^2+y^2). (Key point: Special case)

## 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×
- [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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