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Home  >>  TN XII Math  >>  Complex Numbers
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$P$ represents the variable complex number $z$.Find the locus of $P$,if $Im\bigg[\large\frac{2z+1}{iz+1}\bigg]$$=-2$

This is the first part of the multi-part Q8.

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1 Answer

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Toolbox:
  • If $z=a+ib$ then $\bar{z}=a-ib$.
  • $\mid z\mid=\sqrt{a^2+b^2}$
  • $z^{-1}=\large\frac{a-ib}{a^2+b^2}$
  • $z\bar{z}=a^2+b^2$
  • Also $Re(z)=a,Im(z)=b$
  • If $z_1=a+ib,z_2=c+id$
  • $z_1z_2=(a+ib)(c+id)=(ac-bd)+i(ad+bc)$
  • $\mid z_1z_2\mid=\mid z_1\mid\mid z_2\mid$
Step 1:
Let $z=x+iy$
$\large\frac{2z+1}{iz+1}=\frac{2(x+iy)+1}{i(x+iy)+1}$
$\qquad\quad=\large\frac{2x+1)+2iy}{ix-y+1}$
$\qquad\quad=\large\frac{2x+1)+2iy}{1-y)+ix}\times \big(\large\frac{(1-y)-ix}{(1-y)-ix}\big)$
$\qquad\quad=\large\frac{(2x+1)(1-y)+2xy+i(2y(1-y)-x(2x+1))}{(1-y)^2+x^2}$
Step 2:
$Im\bigg[\large\frac{2z+1}{iz+1}\bigg]$$=-2$
$\large\frac{2y(1-y)-x(2x+1)}{(1-y)^2+x^2}$$=-2$
$2y-2y^2-2x^2-x=-2[1+y^2-2y+x^2]$
$2y-2y^2-2x^2-x=-2-2y^2+4y-2x^2$
$x+2y=2$
The locus of $P$ is a straight line.
answered Jun 10, 2013 by sreemathi.v
 
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