Are you scared of solving questions that involve too much Algebra? Algebra is one of the heavily tested topics in the **CAT exam**. You can expect 4-5 questions from this topic in CAT. A student has to be thorough with this topic to sail through the Quant section. Solve this question from CAT 2017 question paper. Make sure you don't miss the caveat here!

Question 13 : 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

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Let us consider the two ratios given to us.

For (a + 3)^{2} : b^{2} = 9 : 1

⟹ \\frac{(a + 3)^2}{b^2}) = 9

⟹ \\frac{(a + 3)}{b}) = ± 3 ------- (1)

Similarly, we can say that, (a - 1)^{2} : (b - 1)^{2} = 4 : 1,

⟹ \\frac{(a - 1)^2}{(b - 1)^2}) = 4

⟹ \\frac{(a - 1)}{(b - 1)}) = ± 2 ------(2)

From (1), we can say that, a + 3 = ± 3b, So

a = 3b – 3 ------- (3) Or,

a = -3b – 3 ------- (4)

Sub (3) in (2)

⟹ (a – 1) = 2(b – 1) or (a – 1) = -2(b – 1)

⟹ (3b – 4) = 2b – 2 or (3b – 4) = -2b + 2

⟹ b = 2 or b = \\frac{6}{5})

⟹ a = 3 or a = \\frac{3}{5})

Both these cases are not possible since a and b are said to be of opposite signs.

Let’s try condition (4).

Sub (4) in (2), we get

⟹ (a – 1) = 2(b – 1) or (a – 1) = -2(b – 1)

⟹ (-3b – 4) = 2b – 2 or (-3b – 4) = -2b + 2

⟹ b = -\\frac{2}{5}) or b = – 6

⟹ a = -\\frac{9}{5}) or a = 15

Here a = 15 and b = – 6 are possible.

Let’s find a^{2} : b^{2}

⟹ \\frac{a^2}{b^2}) = \\frac{15^2}{-6^2})

⟹ \\frac{a^2}{b^2}) = \\frac{225}{36})

⟹ \\frac{a^2}{b^2}) = \\frac{25}{4})

Hence a^{2} : b^{2} is equal to 25 : 4

The question is **"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 :" **

Choice D is the correct answer.

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