Complex Numbers 2 Question 30
31. Match the statements of Column I with those of Column II.
Here, $z$ takes values in the complex plane and $\operatorname{Im}(z)$ and $\operatorname{Re}(z)$ denote respectively, the imaginary part and
| Column I |  |
|
|---|---|---|
| A. | The set of points $z$ satisfying $|z-i| z||=|z+i| z||$ is contained in or equal to |
|
| B. | The set of points $z$ satisfying $|z+4|+|z-4|=0$ is contained in or equal to |
q. the set of points $z$ satisfying $\operatorname{Im}(z)=0$ |
| C. | If $|w|=2$, then the set of points $z=w-\frac{1}{w}$ is contained in or equal to |
r. the set of points $z$ satisfying $|\operatorname{lm}(z)| \leq 1$ |
| D. | If $|w|=1$, then the set of points $z=w+\frac{1}{w}$ is contained in or equal to |
s. the set of points t. satisfying $|\operatorname{Re}(z)| \leq 2$ the set of points $z$ satisfying $|z| \leq 3$ |
the real part of $z$
(2010)
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Answer:
Correct Answer: 31. True
Solution:
- A. Let $z=x+i y$
$\Rightarrow$ we get $y \sqrt{x^{2}+y^{2}}=0$
$\Rightarrow$
$y=0$
$\Rightarrow$
$I _m(z)=0$
B. We have
$$ \begin{array}{rlrl} \Rightarrow & & 2 a e & =8,2 a=10 \\ \Rightarrow & 10 e & =8 \\ & & e & =\frac{4}{5} \\ \Rightarrow & b^{2} & =25 \quad 1-\frac{16}{25}=9 \\ \therefore & & \frac{x^{2}}{25}+\frac{y^{2}}{9} & =1 \end{array} $$
$w=2(\cos \theta+i \sin \theta)$
$\therefore \quad z=2(\cos \theta+i \sin \theta)-\frac{1}{2(\cos \theta+i \sin \theta)}$
$$ \begin{aligned} & =2(\cos \theta+i \sin \theta)-\frac{1}{2}(\cos \theta-i \sin \theta) \\ & =\frac{3}{2} \cos \theta+\frac{5}{2} i \sin \theta \end{aligned} $$
Let $\quad z=x+i y$
$\Rightarrow \quad x=\frac{3}{2} \cos \theta$ and $y=\frac{5}{2} \sin \theta$
$\Rightarrow \quad \frac{2 x^{2}}{3}+\frac{2 y}{5}^{2}=1$
$\Rightarrow \quad \frac{x^{2}}{9 / 4}+\frac{y^{2}}{25 / 4}=1$
$\therefore \quad e=\sqrt{1-\frac{9 / 4}{25 / 4}}=\frac{4}{5}$
D. Let $\quad w=\cos \theta+i \sin \theta$
Then, $\quad z=x+i y=\cos \theta+i \sin \theta+\frac{1}{\cos \theta+i \sin \theta}$
$$ =2 \cos \theta $$
$\Rightarrow \quad x=2 \cos \theta, y=0$
