# The equation represents a part of a circle having radius equal to :   Option 1) 2 Option 2) 1 Option 3) Option 4)

Using the definition of complex numbers as we studied in Definition of Complex Number -

$z=x+iy, x,y\epsilon R$  & i2=-1

-

Real part of z = Re (z) = x & Imaginary part of z = Im (z) = y

we can say, $\frac{iz-2}{z-i}=\frac{ix-y-2}{x+iy-i}=\frac{-y-2+ix}{x+i(y-1)}$

Proceeding further by Division of Complex Numbers -

$\frac{a+ib}{c+id}=\frac{ac+bd}{c^{2}+d^{2}}+i\frac{bc-ad}{c^{2}+d^{2}}$

We have, $\frac{-yx-2x+xy-x}{x^{2}+(y-1)^{2}}+i\frac{x^{2}+y^{2}+y-2}{x^{2}+(y-1)^{2}}$

It has been given that the Imaginary part of this complex number is -1.

Therefore, $\frac{x^{2}+y^{2}+y-2}{x^{2}+(y-1)^{2}}=-1\Rightarrow 2x^{2}+2y^{2}-y-1=0\Rightarrow x^{2}+y^{2}-(1/2)y-(1/2)=0$

Whose radius is:$\sqrt{\frac{1}{16}+\frac{1}{2}}=\sqrt{\frac{9}{16}}=\frac{3}{4}$

Option 1)

2

This is a wrong option.

Option 2)

1

This is a wrong option.

Option 3)

This is the correct option.

Option 4)

This is a wrong option.

N

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