Question

Let $a, b$ be two real numbers such that a b&lt0; If the complex number $\frac{1+ai}{b+i}$ is of unit modulus and $a$ $+i b$ lies on the circle $|z-I|=|2 z|$, then a possible value of $\frac{1+[a ]}{4 b}$, where $[t]$ is greatest integer function, is :

• A. $\frac{1}{2}$
• B. $-\frac{1}{2}$
• C. $1$
• D. $-1$

Answer 1

Answer: (B)

$-\frac{1}{2}$

Answer 2

ab<0∣∣​b+i1+ai​∣∣​=1
∣1+ai∣=∣b+i∣
a2+1=b2+1⇒a=±b⇒b=−aasab<0
(a,b) lies on ∣z−1∣=∣2z∣
la+ib−1∣=2la+ib∣
(a−1)2+b2=4(a2+b2)
(a−1)2=a2=4(2a2)
1−2a=6a2⇒6a2+2a−1=0
a=12−2±28​​=6−1±7​​
a=67​−1​&b=61−7​​
[a]=0
∴4b1+[a]​=4(1−7​)6​=−(41+7​​)
or [a]=0
Similarly it is not matching with a=6−1−7​​