Question : If $a^{2} + b^{2} + c^{2} = a b + b c + c a,$ then the value of $\frac{a + c}{b}$ is:
Option 1: 3
Option 2: 2
Option 3: 0
Option 4: 1

Question

Question : If $a + b + c = 0$ , then the value of $\frac{a^{2} + b^{2} + c^{2}}{a b + b c + c a}$ is:

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Answer 1

Correct Answer: 2

Solution : Given: $a^{2} + b^{2} + c^{2} = a b + b c + c a$
⇒ $a^{2} + b^{2} + c^{2} - a b - b c - c a = 0$
⇒ $\frac{1}{2} [(a - b)^{2} + (b - c)^{2} + (c - a)^{2}] = 0$
⇒ $(a - b)^{2} + (b - c)^{2} + (c - a)^{2} = 0$
The square of any real number is always non-negative.
Therefore, we have:
$(a - b)^{2} = 0 \Rightarrow a = b$
$(b - c)^{2} = 0 \Rightarrow b = c$
$(c - a)^{2} = 0 \Rightarrow c = a$
Thus, we have $a = b = c$ . Now, we can calculate $\frac{a + c}{b}$ as follows:
$\frac{a + c}{b} = \frac{a + a}{a} = 2$
Hence, the correct answer is 2.

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Question : If a2+b2+c2=ab+bc+ca, then the value of a+cb is:Option 1: 3Option 2: 2Option 3: 0Option 4: 1

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Question : If $a^{2} + b^{2} + c^{2} = a b + b c + c a,$ then the value of $\frac{a + c}{b}$ is:
Option 1: 3
Option 2: 2
Option 3: 0
Option 4: 1