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Given that,

- $(a+3)^{2} : b^{2} = 9 : 1 \quad \longrightarrow (1)$
- $(a-1)^{2} : (b-1)^{2} = 4 : 1 \quad \longrightarrow (2)$

From equation $(1), \frac{(a+3)^{2}}{b^{2}} = \frac{9}{1}$

$\Rightarrow \left(\frac{a+3}{b}\right)^{2} = \left(\frac{3}{1}\right)^{2}$

$\Rightarrow \frac{a+3}{b} = \frac{3}{1}$

$\Rightarrow a+3 = 3b$

$\Rightarrow a=3b-3 \quad \longrightarrow (3)$

From equation $(2),\frac{(a-1)^{2}}{(b-1)^{2}} = \frac{4}{1}$

$\Rightarrow \left(\frac{a-1}{b-1}\right)^{2} = \left(\frac{2}{1}\right)^{2}$

$\Rightarrow \frac{a-1}{b-1} = \frac{2}{1}$

$\Rightarrow a-1 = 2(b-1)$

$\Rightarrow a - 1 = 2b-2 $

$\Rightarrow a = 2b-2+1 $

$\Rightarrow a = 2b-1 \quad \longrightarrow (4)$

Equate the equation $(3)$ and $(4),$ we get

$3b-3 = 2b-1.$

$\Rightarrow 3b-2b = -1+3$

$\Rightarrow \boxed{b=2}$

On putting the value of $b$ in equation $(3),$ we get

$a=3 (2)-3$

$\Rightarrow a = 6-3$

$\Rightarrow \boxed{a=3}$

$\therefore$ The required ratio, $ a^{2}:b^{2} = 3^{2} : 2^{2} = 9:4.$

Correct Answer $ : \text{A}$