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
$x-3=\lambda$
And
$y^2=5-{\lambda }^2$
${\lambda }^2=5-y^2$
Therefore
$5-y^2=(x-3{)}^2$
$(x-3{)}^2+y^2=5$
Hence
$Z$ represents a circle whose center is at $(3,0)$ with radius $\sqrt{5}$.