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If a2 + b2 + c2 - ab - bc - ca = 0, then a ∶ b ∶ c is:
- a)1 ∶ 2 ∶ 1
- b)2 ∶ 1 ∶ 1
- c)1 ∶ 1 ∶ 2
- d)1 ∶ 1 ∶ 1
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
If a2+ b2+ c2- ab - bc - ca = 0, then a bc is:a)1 2 1b)2 1 1c)1 1...
Understanding the Problem
The equation given is:
a² + b² + c² - ab - bc - ca = 0
This is a symmetric equation involving three variables: a, b, and c. We need to find the values of a, b, and c that satisfy this equation.
Rearranging the Equation
The equation can be rearranged as:
a² + b² + c² = ab + bc + ca
This suggests that the sum of the squares of the variables equals the sum of their pairwise products.
Factoring the Expression
We can rewrite the left side as:
(a - b)² + (b - c)² + (c - a)² = 0
For this sum of squares to equal zero, each individual square must also equal zero:
- (a - b)² = 0
- (b - c)² = 0
- (c - a)² = 0
Conclusions from the Above
This leads us to conclude:
- a - b = 0 → a = b
- b - c = 0 → b = c
- c - a = 0 → c = a
Thus, we find that a, b, and c must all be equal.
Finding the Values
If we set a = b = c = k, where k is any real number, the values can be expressed as:
- a = 1
- b = 1
- c = 1
Thus, the only solution is a = 1, b = 1, c = 1.
Final Answer
The values of a, b, and c are indeed:
- a = 1
- b = 1
- c = 1
Therefore, the correct answer is option 'D': 1, 1, 1.
The equation given is:
a² + b² + c² - ab - bc - ca = 0
This is a symmetric equation involving three variables: a, b, and c. We need to find the values of a, b, and c that satisfy this equation.
Rearranging the Equation
The equation can be rearranged as:
a² + b² + c² = ab + bc + ca
This suggests that the sum of the squares of the variables equals the sum of their pairwise products.
Factoring the Expression
We can rewrite the left side as:
(a - b)² + (b - c)² + (c - a)² = 0
For this sum of squares to equal zero, each individual square must also equal zero:
- (a - b)² = 0
- (b - c)² = 0
- (c - a)² = 0
Conclusions from the Above
This leads us to conclude:
- a - b = 0 → a = b
- b - c = 0 → b = c
- c - a = 0 → c = a
Thus, we find that a, b, and c must all be equal.
Finding the Values
If we set a = b = c = k, where k is any real number, the values can be expressed as:
- a = 1
- b = 1
- c = 1
Thus, the only solution is a = 1, b = 1, c = 1.
Final Answer
The values of a, b, and c are indeed:
- a = 1
- b = 1
- c = 1
Therefore, the correct answer is option 'D': 1, 1, 1.
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Community Answer
If a2+ b2+ c2- ab - bc - ca = 0, then a bc is:a)1 2 1b)2 1 1c)1 1...
Given :
Equation is a2 + b2 + c2 - ab - bc - ca = 0
Solution :
Here we have 3 variables and only one equation .
Note To solve these type of question we assume the value of any 2 variables.
Let assume that b = 1 and C = 1
⇒ a2 + 1 + 1 - a - 1 - a = 0
⇒ a2 + 1 - 2a = 0
⇒ ( a - 1 )2 = 0 [ ∵ ( A+B )2 = A2 + B2 + 2AB ]
Now a = 1
Now we have a = 1 , b = 1 and c = 1
So a : b : c = 1 : 1 : 1
Hence the correct answer is "1 : 1 : 1".
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