# CAT 2017 Question Paper | Quants Slot 1

###### CAT Previous Year Paper | CAT Algebra Questions | Question 13

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Question 13 : If a and b are integers of opposite signs such that (a + 3)2 : b2 = 9 : 1 and (a - 1)2 : (b - 1)2 = 4 : 1, then the ratio a2 : b2 is :

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

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##### Method of solving this CAT Question from Algebra

Let us consider the two ratios given to us.
For (a + 3)2 : b2 = 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 a2 : b2 ⟹ $\frac{a^2}{b^2}$ = $\frac{15^2}{-6^2}$ ⟹ $\frac{a^2}{b^2}$ = $\frac{225}{36}$ ⟹ $\frac{a^2}{b^2}$ = $\frac{25}{4}$ Hence a2 : b2 is equal to 25 : 4 The question is "If a and b are integers of opposite signs such that$a + 3)2 : b2 = 9 : 1 and (a - 1)2 : (b - 1)2 = 4 : 1, then the ratio a2 : b2 is :"

##### Hence, the answer is 25 : 4

Choice D is the correct answer.

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