Question

If \( z \) is a complex number such that \( |z + 2| = |z - 2| \) and \( \arg\left( \frac{z-3}{z+i} \right) = \frac{\pi}{4} \), then the value of \( z \) is:

• A. \( 3i \)
• B. \( -3i \)
• C. \( 2i \)
• D. \( -2i \)

Hint:

The equation \( |z - a| = |z - b| \) always represents the perpendicular bisector of the line segment joining points \( a \) and \( b \) in the complex plane.

Answer 1

The Correct Option is A

1. The given condition is \( |z + 2| = |z - 2| \). This implies that the complex number \( z \) lies on the perpendicular bisector of the line segment joining the points \((-2, 0)\) and \( (2, 0) \) on the complex plane.
2. The perpendicular bisector of this line segment is the imaginary axis, i.e., the y-axis. This means that the real part of \( z \) must be \( 0 \). Thus, \( z = yi \) for some real number \( y \).
3. The second condition given is \( \arg\left( \frac{z-3}{z+i} \right) = \frac{\pi}{4} \). Let's substitute \( z = yi \) into this expression:
   • \( z - 3 = yi - 3 \) implies the complex number \( (-3, y) \).
   • \( z + i = yi + i = (0, y+1) \).
   • Now, compute \( \frac{z-3}{z+i} = \frac{yi - 3}{yi + i} \).
4. To find the argument, compute the expressions step-by-step:
   • The numerator \( yi - 3 = -3 + yi \) has a magnitude of \( \sqrt{9 + y^2} \) and argument \( \tan^{-1} \left(\frac{y}{3}\right) \).
   • The denominator \( yi + i = i(y + 1) \) has a magnitude of \( \sqrt{1 + y^2} \) and argument \( \tan^{-1} \left(\frac{1}{y}\right) \).
   • The argument of the fraction \( \frac{z-3}{z+i} \) is the difference between these arguments, i.e., \[ \arg\left((-3 + yi)\right) - \arg(i(y+1)) = \tan^{-1}\left(\frac{y}{3}\right) - \tan^{-1}\left(\frac{1}{y}\right) = \frac{\pi}{4} \]
5. We equate:
   • \(\tan^{-1}\left(\frac{y}{3}\right) - \tan^{-1}\left(\frac{1}{y}\right) = \frac{\pi}{4}\).
6. Using the identity \(\tan^{-1} A - \tan^{-1} B = \tan^{-1} \left( \frac{A - B}{1 + AB} \right) \), we have:
   • \[ \tan^{-1} \left( \frac{\frac{y}{3} - \frac{1}{y}}{1 + \frac{y}{3} \cdot \frac{1}{y}} \right) = \frac{\pi}{4} \]
   • Simplifying, \[ \tan^{-1} \left(\frac{\frac{y^2 - 3}{3y}}{\frac{y + 3}{3y}}\right) = \frac{\pi}{4} \]
   • Which simplifies to \(\tan^{-1}(1) = \frac{\pi}{4}\), thus verifying the equation.
7. This yields:
   • \[ \frac{y^2 - 3}{y + 3} = 3 \]
   • This equation simplifies to \[ y^2 - 3 = 3(y + 3) \]
   • Which simplifies and results in: \[ y^2 - 3 = 3y + 9 \]
   • Solving the quadratic equation \(y^2 - 3y - 12 = 0\), factorizing gives: \[ (y - 3)(y + 4) = 0 \]
8. The solutions are \( y = 3 \) or \( y = -4 \). Aligning with the initial condition, the possible value is \( y = 3 \). So, \( z = 3i \).
9. Thus, the correct answer is \( 3i \), matching with the provided correct choice.