Question

If $P$ represents the variable complex number $z$ and if $|2z-1|=2\left|z\right|$ then the locus of $P$ is:

• A. The straight line $x=\frac{1}{4}$
• B. The straight line $y=\frac{1}{4}$
• C. The straight line $z=\frac{1}{2}$
• D. The circle $x^2+y^2-4x-1=0$

Answer 1

Answer: (A)

The straight line $x=\frac{1}{4}$

Let $z=x+iy$

Given $|2z-1|=2|z|$

$\Rightarrow |2(x+iy)-1|=2|x+iy|$

$\Rightarrow |2x-1+i2y|=2|x+iy|$

$\Rightarrow \sqrt{(2x-1{)}^2+(2y{)}^2}=2\sqrt{x^2+y^2}$

Squaring on both sides we get

$(2x-1{)}^2+4y^2=4(x^2+y^2)$

$\Rightarrow 4x^2-4x+1+4y^2=4x^2+4y^2$

$\Rightarrow -4x+1=0$

$\Rightarrow 4x=1$

$\Rightarrow x=\frac{1}{4}$

Thus if $P$ represents the variable complex number $z$ and if $|2z-1|=2|z|$ then the locus of $P$is the straight line $x=\frac{1}{4}$