The value of λ for which the loci argz = (π)/(6) and | z - 2 3 i |...

z = x + iy$</p><p>$argz = \tan^{- 1} \left(\dfrac{y}{x}\right)

$arg \left( z\right) = \dfrac{\pi}{6}$

$∴ \tan^{- 1}\dfrac{y}{x} = \dfrac{\pi}{6}</p><p>\dfrac{y}{x} =$ tan$\dfrac{\pi}{6} = \dfrac{1}{3}</p><p>$⇒ x = $\sqrt{3}$ y$</p><p>Also given, </p><p>$| z - 2$\sqrt{3}$ i | = \lambda$</p><p>$| x + iy - 2$\sqrt{3}$ i | = \lambda$</p><p>$x^{2} + \left( y - 2\sqrt{3}\right)2 = {λ}^{2}

This is the equation of circle whose centre $x - \sqrt{3}$ y = 0$is$( 0, 2$\sqrt{3} )$ and r = \lambda$. Now, the line touches the circle</p><p>$∴ \lambda = \left|\dfrac{0 - \sqrt{3}\left(2\sqrt{3}\right)}{1 + 3}\right|

$\lambda = \left|\dfrac{6}{2}\right|$ =3$</p><p>$∴ \lambda = 3$</p>$