Question

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

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

Answer 1

The correct option is B Semicircle

Let
$z=x+iy$
$=\lambda +3-i\sqrt{5-{\lambda }^2}$
Hence
$x=\lambda +3$
$(x-3{)}^2={\lambda }^2$
And $y=-\sqrt{5-{\lambda }^2}$
$y^2=5-{\lambda }^2$
$5-y^2={\lambda }^2$
Hence
$5-y^2=(x-3{)}^2$
$(x-3{)}^2+y^2=5$

$y$ is always negative. Hence, the locus is a semi-circle