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Quantitative Aptitude Question on Ratio and Proportion

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

A

9:4

B

81:4

C

1: 4

D

25: 4

Answer

25: 4

Explanation

Solution

(a+3b)2=9 and(a−1b−1)2=4\left(\frac{a+3}{b}\right)^2=9\ \text{and} \left(\frac{a-1}{b-1}\right)^2=4
We have four cases:
Case I : a + 3 = 3b
a – 1 = 2b – 2
Case II : a +3 = 3b
a – 1 = –2b + 2
Case III: a+3 = –3b
a – 1 = 2b – 2
Case IV : a + 3 = –3b
a – 1 = –2b + 2
four cases
I⇒b=2,a=3I ⇒ b=2, a=3 (Rejected)
II,III⇒bII, III ⇒ b is not an integer. (Rejected)
IV⇒b=–6,a=15IV ⇒ b = –6 , a = 15
So,a2b2=(156)2So,\frac{a^2}{b^2}=\left(\frac{15}{6}\right)^2

=254=\frac{25}{4}