Question

Write the correct letter from column (2) against column (1):
$C_1$
(a)$|z-(1+2i)|=5$
(b)$Re\left(\frac{1}{z}\right)<\frac{1}{2}$

(c) $Arg(z-a)=\pi /4,a\in R$

$C_2$
(i)Straight line
(ii) Circle
(iii) Exterior of a circle

Answer 1

Ans. (a)$\to$(ii), (b)$\to$(iii), (c)$\to$(i)
(b)$ \dfrac { 1 }{ z } =\dfrac { 1 }{ x+iy } =\dfrac { x-iy }{ { x
}^{ 2 }+y^{ 2 } }$
$ \therefore Re\left( \dfrac { 1 }{ z } \right) <\dfrac { 1 }{ 2
} \Rightarrow \dfrac { x }{ { x }^{ 2 }+{ y }^{ 2 } } <\dfrac { 1
}{ 2 } $
or $2x<x^2+y^2$ or $ { x }^{ 2 }+{ y }^{ 2
}-2x>0$
or ${(x-1)}^2+y^2>1$
Above represents the exterior of a circle of radius $1$ centred
at $(1,0)$.
(a )Clearing represents a circle centred at $(1,2)$ and of (a) Clearing represents a circle centred at $(1,2)$ and of
radius $5$ as ${(x-1)}^2+(y-2{)}^2=25$
(c) $ \tan { ^{ -1 }\dfrac { y }{ x-a } =\dfrac { \pi }{ 4 } }
\therefore y=(x-a).\tan { =\dfrac { \pi }{ 4 } } $
or $x-y=a$ i.e a St. line