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Q.
If $z=(\lambda+3)-i \sqrt{5-\lambda^{2}}$, then the locus of $z$ is a
Complex Numbers and Quadratic Equations
Solution:
Let $z=x+i y$.
Then $x=\lambda+3$ and $y=\sqrt{5-\lambda^{2}}$
$\Rightarrow (x-3)^{2}=\lambda^{2}$(1)
and $y^{2}=5-\lambda^{2}$(2)
From (1) and (2), $(x-3)^{2}=5-y^{2} $
$\Rightarrow(x-3)^{2}+y^{2}=5$.
Obviously it is a circle.