Question : If $a+b+c=9$ (where $a,b,c$ are real numbers), the minimum value of $a^2+b^2+c^2$ is:
Option 1: 100
Option 2: 9
Option 3: 27
Option 4: 81
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Correct Answer: 27
Solution :
Given: If $a+b+c=9$ (where $a,b,c$ are real numbers).
For the minimum value,
$a=b=c$
⇒ $3a = 9$
⇒ $a = \frac{9}{3}=3$
Similarly, $b=3$ and $c=3$
Replacing $a$, $b$, and $c$ with their values, the equation, $a+b+c=9$, is satisfied.
So, the minimum value of ($a^{2}+b^{2}+c^{2}$) is $3^{2}+3^{2}+3^{2}$ = 27.
Hence, the correct answer is 27.
Related Questions
Question : If $a+b+c=5$ and $a^2+b^2+c^2=15$, then find the value of $a^3+b^3+c^3-3 a b c-27$.
Question : If a, b, c are real and $a^{2}+b^{2}+c^{2}=2(a-b-c)-3, $ then the value of $2a-3b+4c$ is:
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