Consider three observations \( a, b \) and \( c \) such that \( b=a...

Consider three observations \( a, b \) and \( c \) such that \( b=a+c \). If the standard deviation of \( a+2, b+2, c+2 \) is \( d \),
\( \mathrm{P} \) then which of the following is true?

W
[JEE Main-2021]
(a) \( b^{2}=3\left(a^{2}+c^{2}\right)+9 d^{2} \)
(b) \( b^{2}=a^{2}+c^{2}+3 d^{2} \)
(c) \( b^{2}=3\left(a^{2}+c^{2}+d^{2}\right) \)
(d) \( b^{2}=3\left(a^{2}+c^{2}\right)-9 d^{2} \)