If a^2 + b^2 + c^2 - 1024ab - bc - ca = - 2 and a + b = 5c, where c > 0, then the value of c is . . . . . . . .

Answer: Option A

a + b = 5c
$$\frac{a^2+b^2+c^2-1024}{ab-bc-ca}=-2$$
a² + b² + c² + 2ab - 2bc - 2ac = 1024
(a + b)² + c(c - 2b - 2a) = 1024
(5c)² + c[c - 2(b + a)] = 1024
25c² + c[c - 2 × 5c) = 1024
25c² - 9c² = 1024
16c² = 1024
c² = 64
c = 8