Question

Let $a, b$ be two real numbers such that $a b<$ 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

The Correct Option is B

$ab<0\left|\frac{1+ai}{b+i}\right|=1$
$|1+ai|=|b+i|$
$a^2+1=b^2+1\Rightarrow a=\pm b\Rightarrow b=-aasab<0$
$(a,b)\text{lies on}|z-1|=|2z|$
$la+ib-1|=2la+ib|$
$(a-1{)}^2+b^2=4\left(a^2+b^2\right)$
$(a-1{)}^2=a^2=4\left(2a^2\right)$
$1-2a=6a^2\Rightarrow 6a^2+2a-1=0$
$a=\frac{-2\pm \sqrt{28}}{12}=\frac{-1\pm \sqrt{7}}{6}$
$a=\frac{\sqrt{7}-1}{6}\&b=\frac{1-\sqrt{7}}{6}$
$[a]=0$
$\therefore \frac{1+[a]}{4b}=\frac{6}{4(1-\sqrt{7})}=-\left(\frac{1+\sqrt{7}}{4}\right)$
or $[a]=0$
Similarly it is not matching with $a=\frac{-1-\sqrt{7}}{6}$

Answer 2

Provided conditions are:
1. ab < 0,
2. 1 + ai / b + i = 1,
3. a + ib lies on the circle |z − 1| = |2z|.

Step 1: Simplify the Unit Modulus Condition

For 1 + ai / b + i = 1, we have:

|1 + ai| = |b + i|.

Squaring both sides:

a² + 1 = b² + 1.

a² = b².

a = ±b.

Given ab < 0, we deduce: b = −a.

Step 2: Simplify the Circle Condition

The point a + ib lies on the circle |z − 1| = |2z|. Substituting z = a + ib, we have:

|a + ib − 1| = |2(a + ib)|.

Simplify each term:

|a + ib − 1| = |(a − 1) + ib| = √[(a − 1)² + b²].

|2(a + ib)| = 2|a + ib| = 2√(a² + b²).

Equating the two:

√[(a − 1)² + b²] = 2√(a² + b²).

Squaring both sides:

(a − 1)² + b² = 4(a² + b²).

Substitute b = −a:

(a − 1)² + (−a)² = 4(a² + (−a)²).

(a − 1)² + a² = 8a².

a² − 2a + 1 + a² = 8a².

2a² − 2a + 1 = 8a².

6a² + 2a − 1 = 0. (1)

Step 3: Solve for a and b

Solve the quadratic equation 6a² + 2a − 1 = 0 using the quadratic formula:

a = −2 ± √(2² − 4(6)(−1)) / 2(6).

a = −2 ± √(4 + 24) / 12.

a = −2 ± √28 / 12.

a = −2 ± 2√7 / 12.

a = −1 ± √7 / 6.

Since b = −a, we have:

b = 1 ∓ √7 / 6.

Step 4: Calculate 1 + ⌊a⌋ 4b

Substitute a = −1 + √7 / 6:

⌊a⌋ = 0 (since −1 < a < 0).

1 + ⌊a⌋ 4b = 1 / 4b.

Substitute b = 1 − √7 / 6:

1 / 4b = 1 / 4 · 1 − √7 / 6 = 6 / 4(1 − √7).

Rationalize the denominator:

6 / 4(1 − √7) · (1 + √7) / (1 + √7) = 6(1 + √7) / 4(1 − 7) = 6(1 + √7) / −24.

1 / 4b = −1 + √7 / 4.

For a = −1 − √7 / 6, a similar calculation shows that no option matches the result.

Conclusive Answer: No option matches the calculated values. This question was marked as dropped by NTA.