Question

A complex number $z$ satisfying the equation $z=(\alpha +3)+i\sqrt{5-{\alpha }^2}$. Then find the locus of the complex number $z$.

Answer 1

Given, $z$ satisfying the equation $z=(\alpha +3)+i\sqrt{5-{\alpha }^2}$.

or, $x+iy=$$(\alpha +3)+i\sqrt{5-{\alpha }^2}$

or, $x=\alpha +3$ and $y=\sqrt{5-{\alpha }^2}$

or, $\alpha =x-3$.......(1) and $y^2+{\alpha }^2=5$.....(2).

Using (1) in (2) we get,

$y^2+(x-3{)}^2=5$.

This represents a circle whose centre is at $(3,0)$ and radius is $\sqrt{5}$ units.