---
title: "Cylinder with generators parallel to a line"
subject: "Mathematics"
chapter: 3
key: "semester_2/mathematics:cylinder-generators-parallel-line"
source: https://www.bctengineer.com/q/semester_2/mathematics/cylinder-generators-parallel-line
site: BCT Engineering
---

# Cylinder with generators parallel to a line

Mathematics · Chapter 3: Three Dimensional Solid Geometry

- Asked in: 2071 Bhadra, 5 marks
- Video: [Finding equation to the cylinder | #analytical solid geometry | SEM-6 | UNIT-3](https://www.youtube.com/watch?v=mBlCvM0jEMg) — VSP UNITY
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/cylinder-generators-parallel-line
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-3

## Question

Find the equation of the cylinder whose generators are parallel to the line x = −y/2 = z/3 and whose guiding curve is x²+2y² = 1, z = 3.

## Short answer

Generators direction (1,-2,3). Using condition that generator through (x,y,z) meets z=3 on x^2+2y^2=1 gives (3x-z+3)^2+2(3y+2z-6)^2=9.

## Answer

**Given.** Generators parallel to x = −y/2 = z/3, i.e. x/1 = y/(−2) = z/3, so direction ratios (1, −2, 3). Guiding curve: x² + 2y² = 1, z = 3.

#### Generator through P(x, y, z)
The line through P(x₁, y₁, z₁) with direction (1, −2, 3) has points (x₁ + k, y₁ − 2k, z₁ + 3k). It meets the plane z = 3 when z₁ + 3k = 3, i.e. k = (3 − z₁)/3.

The meeting point is

```text
X = x₁ + (3 − z₁)/3 = (3x₁ − z₁ + 3)/3
Y = y₁ − 2(3 − z₁)/3 = (3y₁ + 2z₁ − 6)/3
```

#### On the guiding curve
X² + 2Y² = 1:

```text
(3x₁ − z₁ + 3)²/9 + 2(3y₁ + 2z₁ − 6)²/9 = 1
(3x₁ − z₁ + 3)² + 2(3y₁ + 2z₁ − 6)² = 9
```

#### Locus
Dropping the subscripts:

**(3x − z + 3)² + 2(3y + 2z − 6)² = 9**

Expanded: 9x² + 18y² + 9z² − 6xz + 24yz + 18x − 72y − 54z + 72 = 0.

Check: at z = 3 this gives 9x² + 18y² − 9 = 0 after simplification, i.e. x² + 2y² = 1 ✓.

## Check yourself

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

1. The generators are parallel to x = -y/2 = z/3. What are the direction ratios?
   A. (1, 2, 3)
   B. (1, -2, 3)
   C. (1, -1/2, 3)
   D. (0, -2, 3)
   Answer: B. x/1 = y/(-2) = z/3 gives (1, -2, 3). (Key point: Direction ratios)

2. What is the strategy to find the cylinder equation?
   A. Substitute z = 3 directly into the curve equation
   B. Rotate the guiding curve about the line
   C. Take a point P, draw the generator through it, find where it meets the guiding curve's plane, and impose the curve equation
   D. Find the distance of P from the line and set it equal to 1
   Answer: C. The generator through P meets z = 3 at a point that must lie on x^2+2y^2 = 1. (Key point: Method)

3. The generator through (x1, y1, z1) is (x1+k, y1-2k, z1+3k). For which k does it meet z = 3?
   A. k = (3 - z1)/3
   B. k = (z1 - 3)/3
   C. k = 3 - z1
   D. k = z1/3
   Answer: A. z1 + 3k = 3 gives k = (3 - z1)/3. (Key point: Meeting point)

4. What is the cylinder?
   A. (3x + z - 3)^2 + 2(3y - 2z + 6)^2 = 9
   B. (3x - z + 3)^2 + 2(3y + 2z - 6)^2 = 9
   C. x^2 + 2y^2 = 1
   D. (3x - z + 3)^2 + 2(3y + 2z - 6)^2 = 1
   Answer: B. Substituting X = (3x-z+3)/3, Y = (3y+2z-6)/3 into X^2+2Y^2 = 1 and multiplying by 9 gives this. (Key point: Final locus)

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