Question : If $a^2+b^2+c^2+84 = 4(a - 2b + 4c)$, then $\sqrt{ab - bc + ca}$ is equal to:
Option 1: $4 \sqrt{10}$
Option 2: $\sqrt{10}$
Option 3: $5\sqrt{10}$
Option 4: $2\sqrt{10}$
Correct Answer: $2\sqrt{10}$
Solution :
Given: $a^2+b^2+c^2+84 = 4(a - 2b + 4c)$
⇒ $a^2+ b^2+ c^2− 4a + 8b − 16c + 84 = 0$
⇒ $(a^2− 4a + 4) + (b^2+ 8b + 16) + (c^2− 16c + 64) = 0$
⇒ $(a − 2)^2+(b + 4)^2+ (c − 8)^2= 0$
⇒ $a = 2, b = − 4$, and $c = 8$
Now, $\sqrt{ab - bc + ca}$
= $\sqrt{(2× -4) – (- 4 × 8 ) + (8 × 2)}$
= $\sqrt{-8 +32 + 16}$
= $\sqrt{40}$
= $2\sqrt{10}$
Hence, the correct answer is $2\sqrt{10}$.
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