---
title: "Extreme value of x²+y²+z² with ax+by+cz=p"
subject: "Mathematics"
chapter: 1
key: "semester_2/mathematics:extreme-value-x2-y2-z2"
source: https://www.bctengineer.com/q/semester_2/mathematics/extreme-value-x2-y2-z2
site: BCT Engineering
---

# Extreme value of x²+y²+z² with ax+by+cz=p

Mathematics · Chapter 1: Calculus of Two or more variables

- Asked in: 2073 Bhadra, 5 marks; 2070 Bhadra, 5 marks; 2067 Mangsir, 5 marks (asked 3 times)
- Video: [Find the minimum value of x^2+y^2+z^2 given that ax+by+cz=p](https://www.youtube.com/watch?v=wiivltEFPTY) — SamalsPhysics💞 
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/extreme-value-x2-y2-z2
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-1

## Question

Find the extreme value of x²+y²+z² connected by the relation ax+by+cz = p.

## Short answer

Applying Lagrange multipliers to x²+y²+z² with constraint ax+by+cz=p gives x=ap/(a²+b²+c²), y=bp/(…), z=cp/(…). The extreme value is the minimum distance squared from origin to the plane: p²/(a²+b²+c²).

## Answer

Find the stationary value of f = x² + y² + z² subject to g = ax + by + cz − p = 0, using Lagrange's multiplier λ.

#### Setting up
F = x² + y² + z² + λ(ax + by + cz − p)

```text
∂F/∂x = 2x + λa = 0   →  x = −λa/2
∂F/∂y = 2y + λb = 0   →  y = −λb/2
∂F/∂z = 2z + λc = 0   →  z = −λc/2
```

#### Finding λ
Put these into the constraint ax + by + cz = p:

−(λ/2)(a² + b² + c²) = p, so λ = −2p/(a² + b² + c²).

Then

```text
x = ap/(a² + b² + c²)
y = bp/(a² + b² + c²)
z = cp/(a² + b² + c²)
```

#### Extreme value
```text
x² + y² + z² = p²(a² + b² + c²)/(a² + b² + c²)²
             = p²/(a² + b² + c²)
```

#### Nature
x² + y² + z² is the squared distance of a point of the plane ax + by + cz = p from the origin. A plane has a nearest point but no farthest point, so the value is a minimum.

**Answer: extreme (minimum) value = p²/(a² + b² + c²).**

## Check yourself

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

1. For F = x²+y²+z² + λ(ax+by+cz−p), what does ∂F/∂x = 0 give?
   A. x = λa
   B. x = −λa/2
   C. x = −λ/(2a)
   D. x = 2λa
   Answer: B. 2x + λa = 0, so x = -λa/2; similarly for y and z. (Key point: Stationary conditions)

2. Substituting into ax+by+cz = p gives which λ?
   A. λ = −2p/(a+b+c)
   B. λ = p/(a²+b²+c²)
   C. λ = 2p/(a²+b²+c²)
   D. λ = −2p/(a²+b²+c²)
   Answer: D. -(λ/2)(a²+b²+c²) = p, so λ = -2p/(a²+b²+c²). (Key point: Finding λ)

3. What is the extreme value of x²+y²+z²?
   A. p²/(a+b+c)²
   B. p/(a²+b²+c²)
   C. p²(a²+b²+c²)
   D. p²/(a²+b²+c²)
   Answer: D. Using x = ap/S etc. with S = a²+b²+c², the sum is p²S/S² = p²/S. (Key point: Extreme value)

4. Why is the extreme value a minimum?
   A. It is the squared distance from the origin to a point of the plane; a plane has a nearest but no farthest point
   B. Because a, b, c are positive
   C. Because λ is negative
   D. Because p² is positive
   Answer: A. x²+y²+z² is the squared distance to the origin; on a plane this has a minimum but no maximum. (Key point: Nature)

## Close variants set in other exams

- [Extreme value of x²+y²+z² with x+y+z=3a](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aextreme-value-x2-y2-z2-2) — Mathematics, chapter 1, asked 5×
- [Extreme value with two linear constraints](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aextreme-value-two-linear-constraints) — Mathematics, chapter 1, asked 5×

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