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 \(a:b\) is

• A. 9:4
• B. 81:4
• C. 1: 4
• D. 25: 4

Answer 1

The Correct Option is D

\(\left(\frac{a+3}{b}\right)^2=9\ \text{and} \left(\frac{a-1}{b-1}\right)^2=4\)
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

\(I ⇒ b=2, a=3\) (Rejected)
\( II, III ⇒ b\) is not an integer. (Rejected)
\(IV ⇒ b = –6 , a = 15\)
\(So,\frac{a^2}{b^2}=\left(\frac{15}{6}\right)^2\)

\(=\frac{25}{4}\)