Question

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

• A. circle
• B. straight line
• C. parabola
• D. None of these

Answer 1

The correct option is A

circle

Explanation for the correct option:

Find the locus of $z$.

Compare the real part of $z=\left(\lambda +3\right)+i\sqrt{5-{\lambda }^2}$ with $z=x+iy$.

$$\begin{gathered} \lambda +3=x \\ \Rightarrow \lambda =x-3...\left(1\right) \end{gathered}$$

Now compare the imaginary part.

$y=\sqrt{5-{\lambda }^2}$

Now square both sides and use the equation $1$ to substitute $x-3$ for $\lambda$.

$$\begin{gathered} y^2=\left(\sqrt{5-{\lambda }^2}\right) \\ \Rightarrow y^2=5-{\left(x-3\right)}^2\left[\mathrm{Using}\mathrm{equation}1\right] \\ \Rightarrow {\left(x-3\right)}^2+y^2={\left(\sqrt{5}\right)}^2 \end{gathered}$$

So this is an equation of circle whose center is $\left(3,0\right)$ and radius is $\sqrt{5}$.

The locus of $z$ is a circle.

Hence, the correct option is A.