CAT 2017 Slot 1 QA Question & Solution
Question
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
Options
Solution
Since the square root can be positive or negative we will get two cases for each of the equation.
For the first one,
a + 3 = 3b ......................... (1)
a + 3 = -3b ......................... (2)
For the second one,
a - 1 = 2(b -1) ......................... (3)
a - 1 = 2 (1 - b) ......................... (4)
we have to solve (1) and (3), (1) and (4), (2) and (3), (2) and (4).
Solving (1) and (3),
a + 3 = 3b and a = 2b - 1, solving, we get a = 3 and b = 2, which is not what we want.
Solving (1) and (4)
a + 3 = 3b and a = 3 - 2b, solving, we get b = 1.2, which is not possible.
Solving (2) and (3)
a + 3 = -3b and a = 2b - 1, solving, we get b = 0.4, which is not possible.
Solving (2) and (4),
a + 3 = -3b and a = 3 - 2b, solving, we get a = 15 and b = -6 which is what we want.
Thus, $\frac{a^2}{b^2} = \frac{25}{4}$