---
title: "Cone with vertex origin and circle base"
subject: "Mathematics"
chapter: 3
key: "semester_2/mathematics:cone-vertex-origin-circle-base"
source: https://www.bctengineer.com/q/semester_2/mathematics/cone-vertex-origin-circle-base
site: BCT Engineering
---

# Cone with vertex origin and circle base

Mathematics · Chapter 3: Three Dimensional Solid Geometry

- Asked in: 2073 Magh, 5 marks; 2071 Bhadra, 5 marks (asked 2 times)
- Video: [Engineering mathematics | Episode 4 | Cone | All old question solution](https://www.youtube.com/watch?v=EETi2ndndSE) — Sudip Dhakal
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/cone-vertex-origin-circle-base
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-3

## Question

Find the equation of the cone whose vertex is the origin and base the circle y²+z² = b², x = a.

## Short answer

The cone with vertex at origin and base circle y²+z²=b², x=a is a²(y²+z²) = b² x².

## Answer

**Cone.** Vertex O(0, 0, 0); base (guiding curve) the circle y² + z² = b², x = a.

#### Generator
A line through the origin with direction ratios (l, m, n): x/l = y/m = z/n.   ... (1)

It meets the plane x = a at the point where x/l = a/l = t, so

(a, a m/l, a n/l).

This point lies on the circle y² + z² = b²:

```text
(a m/l)² + (a n/l)² = b²
a² (m² + n²) = b² l²        ... (2)
```

#### Eliminate l, m, n
From (1), m/l = y/x and n/l = z/x, i.e. m : n : l = y : z : x. Put (l, m, n) = (x, y, z) in (2):

```text
a² (y² + z²) = b² x²
```

**Answer: the cone is a²(y² + z²) = b² x²**, a right circular cone with axis along the x-axis.

## Check yourself

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

1. Where does the line x/l = y/m = z/n meet x = a?
   A. At (a, l/m, l/n)
   B. At (a, am, an)
   C. At (a, am/l, an/l)
   D. At (a, a l/m, a l/n)
   Answer: C. x/l = a/l gives y = am/l and z = an/l. (Key point: Generator meets plane)

2. What relation do l, m, n satisfy?
   A. a^2(m^2 + n^2) = b^2 l^2
   B. a^2(m^2 + n^2) = b^2 l
   C. a(m^2 + n^2) = b^2 l^2
   D. (m^2 + n^2) = b^2 l^2
   Answer: A. Substituting into y^2 + z^2 = b^2 and clearing l^2 gives this. (Key point: Using the base)

3. How are l, m, n eliminated?
   A. Replace (l, m, n) by (a, b, 0)
   B. Replace (l, m, n) by (x, y, z), since the generator passes through the origin
   C. Replace l by a and m, n by b
   D. Replace (l, m, n) by (1, 1, 1)
   Answer: B. Direction ratios of OP are proportional to its coordinates. (Key point: Elimination)

4. What is the cone and its type?
   A. a^2(y^2 + z^2) = b^2 x^2, a right circular cone about the x-axis
   B. b^2(y^2 + z^2) = a^2 x^2, about the x-axis
   C. a^2(y^2 + z^2) = b^2 x^2, about the y-axis
   D. a^2(y^2 + z^2) = b^2 x, about the x-axis
   Answer: A. The base is a circle centred on the x-axis, so the cone is right circular about the x-axis. (Key point: Final cone)

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