If the least and the largest real values of α, for which the...

Put $z = x + iy$

$x + iy + \alpha\left|x + iy - 1\right| + 2i = 0$

$⇒ x + \alpha\sqrt{{\left(x - 1\right)}^{2} + {y}^{2}} + i\left(y + 2\right) = 0 + 0i$

$⇒ y + 2 = 0$ and $x + \alpha\sqrt{{\left(x - 1\right)}^{2} + {y}^{2}} = 0$

$⇒ y = - 2$ and ${\alpha}^{2} = \dfrac{{x}^{2}}{{x}^{2} - 2x + 5}$

Now $\dfrac{{x}^{2}}{{x}^{2} - 2x + 5} \in \left[0, \dfrac{5}{4}\right]$

$∴ {\alpha}^{2} \in \left[0, \dfrac{5}{4}\right] ⇒ \alpha \in \left[- \dfrac{\sqrt{5}}{2}, \dfrac{\sqrt{5}}{2}\right]$

$∴ p = - \dfrac{\sqrt{5}}{2}; q = \dfrac{\sqrt{5}}{2}$

$⇒ 4\left({p}^{2} + {q}^{2}\right) = 4\left(\dfrac{5}{4} + \dfrac{5}{4}\right) = 10$