Here ∣z1−3∣=21 represents a circle on argand plane with centre (3,0) and radius 21
Given ∣z2+∣z2−1∣∣2=∣z2−∣z2+1∣∣2
⇒(z2+∣z2−1∣)(z¯2+∣z2−1∣)=(z2−∣z2+1∣)(z¯2−∣z2+1∣)
⇒z2(∣z2−1∣+∣z2+1∣)+z¯2(∣z2−1∣+∣z2+1∣)=∣z2+1∣2−∣z2−1∣2
⇒(z2+z¯2)(∣z2+1∣+∣z2−1∣)=2(z2+z¯2)
⇒ Either z2+z¯2=0 or \left|{z}_{2} + 1\right| + \left|{z}_{2} - 1\right| = 2</p><p>i.e.{z}_{2}liesonimaginaryaxisoritliesonthelinesegmentjoining(−1,0) and \left(1, 0\right)</p><p><imgsrc="https://cdn−question−pool.getmarks.app/pyq/jeemain/cfa52ac5−9902−4fd6−99c7−2e86e16274ed−image.png"style="width:315px;height:200px;"/></p><p>So,theminimumdistancebetween{z}_{1} \text{ and } {z}_{2}willbethedistancebetweenthepoints(1,0) and (25,0)
Hence, {\left|{z}_{1} - {z}_{2}\right|}_\min = \dfrac{3}{2}
