Question

If the least and the largest real values of \(\alpha\), for which the equation \(|z| + \alpha(z - 1) + 2i = 0\) (\(z \in C\) and \(i = \sqrt{-1}\)) has a solution, are \(p\) and \(q\) respectively; then \(4(p^2 + q^2)\) is equal to _________

Hint:

For equations involving $|z|$, substitute $z=x+iy$ and resolve into two real equations by comparing $Re(z)$ and $Im(z)$ parts. Ensure that the squared equation's condition (e.g., $1-x \geq 0$ if $\alpha>0$) is satisfied.

Answer 1

Correct Answer: 10

Step 1: Let \(z = x + iy\). Equation: \(\sqrt{x^2+y^2} + \alpha(x + iy - 1) + 2i = 0\).
Step 2: Equate imaginary parts: \(\alpha y + 2 = 0 \Rightarrow y = -2/\alpha\).
Step 3: Equate real parts: \(\sqrt{x^2 + y^2} + \alpha(x-1) = 0 \Rightarrow \sqrt{x^2 + 4/\alpha^2} = \alpha(1-x)\).
Step 4: Squaring: \(x^2 + 4/\alpha^2 = \alpha^2(1+x^2-2x) \Rightarrow (\alpha^2-1)x^2 - 2\alpha^2x + (\alpha^2 - 4/\alpha^2) = 0\).
Step 5: For real \(x\), \(D \geq 0\). Solving \(4\alpha^4 - 4(\alpha^2-1)(\alpha^2 - 4/\alpha^2) \geq 0\) gives \(\alpha^2 \leq 5/4\).
Step 6: \(p = -\sqrt{5}/2, q = \sqrt{5}/2\). \(4(p^2 + q^2) = 4(5/4 + 5/4) = 10\).