if a 2 +b 2 +c 2 -ab-bc-ca=0 then prove that a=b=c

Question

SHAIK AASIF AHAMED · Answer

Hello student,
Please find the answer to your question below
a² + b² + c² = ab + bc + ca
Multiplying both sides with "2", we have
2 ( a² + b² + c² ) = 2 ( ab + bc + ca)
2a² + 2b² + 2c² = 2ab + 2bc + 2ca
a² + a² + b² + b² + c² + c² - 2ab - 2bc - 2ca = 0
a² + b² - 2ab + b² + c² - 2bc + c² + a² - 2ca = 0
(a² + b² - 2ab) + (b² + c² - 2bc) + (c² + a² - 2ca) = 0
(a - b)² + (b - c)² + (c - a)² = 0
=>
(a - b)² = (b - c)² = (c - a)² = 0
=>
(a - b)² = 0 ---------- (1)
(b - c)² = 0 ---------- (2)
(c - a)² = 0 ---------- (3)
Simplifying Equ. (1), we have
(a - b)² = 0
Taking Square Root on both sides, we have
a - b = 0
a = b ---------- (4)
Simplifying Equ. (2), we have
(b - c)² = 0
Taking Square Root on both sides, we have
b - c = 0
b = c ---------- (5)
Simplifying Equ. (3), we have
(c - a)² = 0
Taking Square Root on both sides, we have
c - a = 0
c = a ---------- (6)
From Equation No. (4), (5) & (6) , it is proved that
a = b = c

Kushagra Madhukar · Answer

Dear student, Please find the attached solution to your problem. As we know, a 3 + b 3 + c 3 – 3abc = ( a + b + c )( a 2 + b 2 + c 2 – ab – bc – ca ) Given, a 2 + b 2 + c 2 – ab – bc – ca = 0 Hence, a 3 + b 3 + c 3 – 3abc = 0 or, (a 3 + b 3 + c 3 )/3 = abc or, (a 3 + b 3 + c 3 )/3 = ( a 3 * b 3 * c 3 ) 1/3 Hence, AM = GM which is only possible when all the terms are equal Hence, a 3 = b 3 = c 3 or, a = b = c Hence proved. Hope it helps. Thanks and regards, Kushagra

Thank you for registering.

Register Now and Win Upto 25% Scholorship for a Full Academic Year !

Enter Your Details

Registration done!

Guest

if a 2 +b 2 +c 2 -ab-bc-ca=0 then prove that a=b=c

if a²+b²+c²-ab-bc-ca=0 then prove that a=b=c

2 Answers

Think You Can Provide A Better Answer ?

Other Related Questions on Algebra

ASK QUESTION

Answer and earn gift vouchers

View courses by askIITians

Design classes One-on-One in your own way with Top IITians/Medical Professionals

Complete Self Study Package designed by Industry Leading Experts

Live 1-1 coding classes to unleash the Creator in your Child

a Complete All-in-One Study package Fully Loaded inside a Tablet!

Register Now & Win Upto 25% Scholorship

Thank you for registering.

Register Now and Win Upto 25% Scholorship for a Full Academic Year !

Enter Your Details

Registration done!

Guest

IIT JEE Courses

One Year IIT Programme

Two Year IIT Programme

Crash Course

NEET Courses

One Year NEET Programme

Two Year NEET Programme

One to One

School Board

Live Online classes

Class 11 & 12

Class 9 & 10

Class 6, 7 & 8

Test Series

JEE test series

NEET test series

C.B.S.E test series

Complete Self Study Packages

FULL COURSE

For class 12th

For class 11th

if a 2 +b 2 +c 2 -ab-bc-ca=0 then prove that a=b=c

if a2+b2+c2-ab-bc-ca=0 then prove that a=b=c

2 Answers

Think You Can Provide A Better Answer ?

Other Related Questions on Algebra

ASK QUESTION

Answer and earn gift vouchers

View courses by askIITians

Design classes One-on-One in your own way with Top IITians/Medical Professionals

Complete Self Study Package designed by Industry Leading Experts

Live 1-1 coding classes to unleash the Creator in your Child

a Complete All-in-One Study package Fully Loaded inside a Tablet!

Register Now & Win Upto 25% Scholorship

if a²+b²+c²-ab-bc-ca=0 then prove that a=b=c