---
title: "Cone with vertex (α,β,γ) and parabola guiding curve"
subject: "Mathematics"
chapter: 3
key: "semester_2/mathematics:cone-vertex-parabola-guiding-curve"
source: https://www.bctengineer.com/q/semester_2/mathematics/cone-vertex-parabola-guiding-curve
site: BCT Engineering
---

# Cone with vertex (α,β,γ) and parabola guiding curve

Mathematics · Chapter 3: Three Dimensional Solid Geometry

- Asked in: 2071 Magh, 5 marks; 2068 Bhadra, 5 marks (asked 2 times)
- Video: [Find the Equation of Conic Whose Vertex is (p,q,r) and Guiding Curve is y^2=4ax, z=0, Geometry B.sc.](https://www.youtube.com/watch?v=DzB_cR4ugtQ) — EDUCATION TRIP
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/cone-vertex-parabola-guiding-curve
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-3

## Question

Find the equation of cone with vertex (α, β, γ) and guiding curve is parabola y² = 4ax, z = 0.

## Short answer

The cone equation is (βz − γy)² = 4a (αz − γx)(z − γ).

## Answer

**Cone.** Vertex V(α, β, γ); guiding curve the parabola y² = 4ax, z = 0.

#### Generator
A line through V with direction ratios (l, m, n):

(x − α)/l = (y − β)/m = (z − γ)/n.   ... (1)

It meets the plane z = 0 where (z − γ)/n = −γ/n. So the point is

```text
x₁ = α − γ l/n,   y₁ = β − γ m/n,   z = 0
```

This point is on the parabola: y₁² = 4a x₁.

```text
(β − γ m/n)² = 4a (α − γ l/n)       ... (2)
```

#### Eliminate l, m, n
From (1), l/n = (x − α)/(z − γ) and m/n = (y − β)/(z − γ).

```text
α − γ l/n = [α(z − γ) − γ(x − α)]/(z − γ) = (αz − γx)/(z − γ)
β − γ m/n = [β(z − γ) − γ(y − β)]/(z − γ) = (βz − γy)/(z − γ)
```

Put in (2):

```text
(βz − γy)²/(z − γ)² = 4a (αz − γx)/(z − γ)
```

**Answer: the cone is (βz − γy)² = 4a (αz − γx)(z − γ).**

Check: at z = 0 this reduces to γ²y² = 4a γ² x, i.e. y² = 4ax.

## Check yourself

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

1. Where does the generator through (alpha, beta, gamma) meet z = 0?
   A. x1 = alpha - gamma l/n, y1 = beta - gamma m/n
   B. x1 = alpha + gamma l/n, y1 = beta + gamma m/n
   C. x1 = alpha - gamma n/l, y1 = beta - gamma n/m
   D. x1 = alpha l/n, y1 = beta m/n
   Answer: A. (z - gamma)/n = -gamma/n at z = 0. (Key point: Intersection with plane)

2. What condition does this point satisfy?
   A. It lies on the plane z = gamma
   B. It lies on the parabola: y1^2 = 4a x1
   C. It lies on the line through the vertex and origin
   D. y1 = 4a x1
   Answer: B. The point must lie on the guiding curve. (Key point: Condition)

3. What are l/n and m/n in terms of x, y, z?
   A. (z - gamma)/(x - alpha) and (z - gamma)/(y - beta)
   B. x/z and y/z
   C. (x - alpha)/(z - gamma) and (y - beta)/(z - gamma)
   D. (x - alpha)/z and (y - beta)/z
   Answer: C. From the generator equations (x - alpha)/l = (z - gamma)/n etc. (Key point: Replacing ratios)

4. What is the cone?
   A. (beta z - gamma y)^2 = 4a(alpha z - gamma x)(z - gamma)
   B. (beta z - gamma y)^2 = 4a(alpha z - gamma x)
   C. (beta z - gamma y) = 4a(alpha z - gamma x)(z - gamma)
   D. (beta z - gamma y)^2 = 4a(alpha z + gamma x)(z - gamma)
   Answer: A. Clearing denominators in (beta z - gamma y)^2/(z-gamma)^2 = 4a(alpha z - gamma x)/(z - gamma). (Key point: Final equation)

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