If $ a^{2}+b^{2}+c^{2}+84 = 4(a-2b+4c),$ then $\sqrt{ab-bc+ca}$ is equal to :

$2\sqrt{10}$

a² + b² + c² + 84 = 4(a - 2b + 4c)

Arrange the equation in this format given below,

= a² + b² + c² + 84 - 4a + 8b -16c = 0

= a² - 4a + 4 +b² + 8b + 16+ c² - 16c + 64 = 0

= (a – 2)² + (b + 4)² + (c-8)² = 0

Now,

a = 2

b = -4

c = 8

$\sqrt{ab-bc+ca}$ = $\sqrt{2 \times -4 - (-4) \times 8 + 8 \times 2}$

$\sqrt{ab-bc+ca}$  = $\sqrt{40}$ = $2\sqrt{10}$