Question : If $a^{2}+13b^{2}+c^{2}-4ab-6bc=0$, then $a:b:c$ is:
Option 1: $1:2:3$
Option 2: $2:3:1$
Option 3: $2:1:3$
Option 4: $1:3:2$
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Correct Answer: $2:1:3$
Solution :
Given: $a^{2} + 13b^{2} + c^{2} - 4ab - 6bc = 0$
⇒ $a^{2} - 4ab + 4b^{2} + 9b^{2} + c^{2} - 6bc = 0$
⇒ $a^{2} - 4ab + 4b^{2} + c^{2} - 6bc + 9b^{2} = 0$
⇒ $(a - 2b)^{2} + (c - 3b)^{2} = 0$
⇒ $a - 2b = 0$ and $c - 3b = 0$ [if the sum of the squares of two numbers is zero then each of the numbers will also be zero]
⇒ $a = 2b$ and $c = 3b$
⇒$a:b = 2: 1$ and $b : c = 1: 3$
$\therefore a:b:c = 2:1:3$
Hence, the correct answer is $a:b:c = 2:1:3$.
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