If a² + b² + c² + 216 = 12(a + b - 2c)

then \(\sqrt {ab-bc-ca}\) is equal to:

6\(\sqrt {5}\)

⇒ a² + b² + c² + 216 = 12(a + b - 2c)

⇒ a² + b² + c² + 216 = 2(6a + 6b - 12c)

⇒ a² + b² + c² - 2(6a) - 2(6b) + 2(12c) + (36 + 36 + 144) = 0

⇒ [a² + 36 - 2(6a)] + [b² + 36 - 2(6b)] + [c² + 144 + 2(12c) ] = 0

⇒ (a - 6)² + (b - 6)² + (c + 12)² = 0

from this we can directly conclude that

a = 6,  b = 6 and c = -12

Now, 

⇒ \(\sqrt {ab - bc - ca}\) = \(\sqrt {(6×6) + (12×6) + (12×6)}\) =\(\sqrt {180}\) = 6\(\sqrt {5}\)