Let in a series of 2n observations, half of them are equal to a and...

Let observations are denoted by $xi$ for $1 \le i < 2n$

$\overset{¯}{x} = \dfrac{∑ {x}_{i}}{2n} = \dfrac{( a + a + \ldots + a ) - ( a + a + \ldots + a )}{2n}$

$⇒ \overset{¯}{x} = 0$

and ${\sigma}_{x}^{2} = \dfrac{∑ {x}_{i}^{2}}{2n} - {\left(\overset{¯}{x}\right)}^{2} = \dfrac{{a}^{2} + {a}^{2} + \ldots + {a}^{2}}{2n} - 0 = {a}^{2}$

$⇒ {\sigma}_{x} = a$

Now, adding a constant $b$ then $\overset{¯}{y} = \overset{¯}{x} + b = 5$

$⇒ b = 5$

and ${\sigma}_{y} = {\sigma}_{x}$ (No change in S.D.) $⇒ a = 20$

$⇒ {a}^{2} + {b}^{2} = 425$