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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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Explanatory Answer

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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