Question

If : \(|z + 1| = αz + ß(i + 1)\) and \(z = \frac 12 - 2i\), find \(α + ß\).

Answer 1

Let the given complex number be:

\[ z = \frac{1}{2} - 2i \]

We are given the equation:

\[ |z + 1| = \alpha z + \beta (1 + i) \]

Substitute the value of \( z \):

\[ \left|\frac{3}{2} - 2i\right| = \frac{\alpha}{2} - 2\alpha i + \beta + \beta i \]

Now, simplify the right-hand side:

\[ \left|\frac{3}{2} - 2i\right| = \left(\frac{\alpha}{2} + \beta\right) + (\beta - 2\alpha)i \]

To satisfy equality between magnitudes and arguments, we compare both sides. From this, we obtain the following relationships:

\[ \beta = 2\alpha \]

\[ \frac{\alpha}{2} + \beta = \sqrt{\left(\frac{9}{4}\right) + 4} \]

Solving these gives:

\[ \alpha + \beta = 3 \]

Final Result:

The relationship between \( \alpha \) and \( \beta \) is: \[ \boxed{\alpha + \beta = 3} \]

Explanation:

• The modulus of a complex number \( |x + yi| \) is given by \( \sqrt{x^2 + y^2} \).
• By comparing the real and imaginary parts, we derive linear relations between \( \alpha \) and \( \beta \).
• Finally, substituting these relations simplifies to \( \alpha + \beta = 3 \).

Hence, the correct option is: (2) 3