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

$9:4$

$81: 4$

$1: 4$

$25: 4$

✅ Correct Option: 4

This question is marked as hard, not because the concept is complex but because it is lengthier than most ratio questions.

When we see $(a+3)^2 : b^2 = 9:1$ , this means:

$\small \dfrac{(a+3)^2}{b^2} = \dfrac{9}{1} = 9$

$\small \dfrac{(a-1)^2}{(b-1)^2} = \dfrac{4}{1} = 4$

$\dfrac{(a+3)^2}{b^2} = 9$

Taking square root of both sides:

$\dfrac{|a+3|}{|b|} = 3$

This gives us: $|a+3| = 3|b|$

Two cases:

Case 1a: $a+3 = 3b$
Case 1b: $a+3 = -3b$

From the second equation:

$\dfrac{(a-1)^2}{(b-1)^2} = 4$

Taking square root:

$\dfrac{|a-1|}{|b-1|} = 2$

This gives us: $|a-1| = 2|b-1|$

Two cases:

Case 2a: $a-1 = 2(b-1) = 2b-2$ , so $a = 2b-1$
Case 2b: $a-1 = -2(b-1) = -2b+2$ , so $a = -2b+3$

The above equations give 4 cases:

Case	Equation 1	Equation 2	Substitution	a	b
1	$a+3 = 3b$	$a = 2b-1$	$(2b-1)+3 = 3b \newline 2b+2 = 3b \newline b = 2$	3	2
2	$a+3 = 3b$	$a = -2b+3$	$(-2b+3)+3 = 3b \newline -2b+6 = 3b \newline 6 = 5b$	0.6	1.2
3	$a+3 = -3b$	$a = 2b-1$	$(2b-1)+3 = -3b \newline 2b+2 = -3b \newline 5b = -2$	-1.8	-0.4
4	$a+3 = -3b$	$a = -2b+3$	$(-2b+3)+3 = -3b \newline -2b+6 = -3b \newline b = -6$	15	-6

Only Case 4 gives integer solutions with opposite signs.

$a^2 : b^2 = 15^2 : (-6)^2$

$= 225 : 36$

$= 25:4$

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