Answer:
(i)
Expanding,
a2 + c2 – 2ac = 4(ab – ac – b2 + bc)
a2 + 4c2b2 + 2ac – 4ab – 4bc = 0
(a + c – 2b)2 = 0
a + c – 2b = 0
a, b, c are in AP
b – a = c – b
a + c – 2b = 0
a + c = 2b
(a – c)2 = 4 (a – b) (b – c)
(ii)
Expanding,
a2 + c2 + 4ac = 2 (ab + bc + ca)
a2 + c2 + 2ac – 2ab – 2bc = 0
(a + c – b)2 – b2 = 0
a + c – b = b
a + c – 2b = 0
2b = a+c
b = (a+c)/2
a, b, c are in AP
b – a = c – b
b = (a+c)/2
a2 + c2 + 4ac = 2 (ab + bc + ca)