Question

The curve represented by $z=\frac{3}{2+\cos \theta +i\sin \theta },\theta \in R$

• A. Never meets the imaginary axis
• B. Meets the real axis in exactly two points
• C. Has maximum value of $|z|$ as $3$
• D. Has minimum value of $|z|$ as $1$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A Never meets the imaginary axis
B Meets the real axis in exactly two points
C Has maximum value of $|z|$ as $3$
D Has minimum value of $|z|$ as $1$

$z=\frac{3}{2+e^{i\theta }}$
$\Rightarrow z=\frac{3(2+e^{-i\theta })}{(2+e^{i\theta })(2+e^{-i\theta })}$
$\Rightarrow z=\frac{6+3e^{-i\theta }}{4+co{s}^2\theta +4cos\theta +si{n}^2\theta }$
$\Rightarrow z=\frac{6+3cos\theta -3isin\theta }{4+co{s}^2\theta +4cos\theta +si{n}^2\theta }$
Thus, $z$ meets the real axis in $2$ points and does not meet the imaginary axis.
Also, $|z|=\frac{3}{\sqrt{4+4cos\theta +co{s}^2\theta +si{n}^2\theta }}=\frac{3}{\sqrt{5+4cos\theta }}$
Thus, the minimum value of $|z|$ will be $\frac{3}{\sqrt{9}}=1$
The maximum value of $|z|$ will be $\frac{3}{\sqrt{1}}=3$
Hence options B, C, D are correct.