Question

If $z=(\lambda +3)-i\sqrt{5-{\lambda }^2}$, where $\lambda$ is real constant, then the locus of $z$ is

• A. Ellipse
• B. Semicircle
• C. Parabola
• D. Straight line

Answer 1

The correct option is B Semicircle

Given : $z=(\lambda +3)-i\sqrt{5-{\lambda }^2}$
Let $z=x+iy$, then
$x=\lambda +3$ and $y=-\sqrt{5-{\lambda }^2}$
$\Rightarrow (x-3{)}^2={\lambda }^2\cdots (1)$
And
$y^2=5-{\lambda }^2\cdots \left(2\right)$
From $\left(1\right)$ and $\left(2\right)$,
$(x-3{)}^2=5-y^2\Rightarrow (x-3{)}^2+y^2=5$
Obviously it is a semicircle as $y<0$. Hence part of the circle lies below the $x-$axis.