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If $a$ and $b$ are integers of opposite signs such that $\left ( a+3 \right )^{2}:b^{2}= 9:1$ and $\left ( a-1 \right )^2 : \left ( b-1 \right )^2= 4:1$ , then the ratio $a^{2}:b^{2}$ is

1. $9:4$
2. $81:4$
3. $1:4$
4. $25: 4$

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 $: \text{A}$

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