MHB Solve Complex Number IV Problem: Find Radius & Centre

[Punch]

A complex number is represented by the point P in an Argand diagram. If the real part of the complex number w=\frac{z+1}{z-2i} (z not 2i) is zero, show that the locus of P is a circle and find the radius and centre of the circle.

I have a problem manipulating w to find the real part of w

 

[Sudharaka]

A complex number is represented by the point P in an Argand diagram. If the real part of the complex number w=\frac{z+1}{z-2i} (z not 2i) is zero, show that the locus of P is a circle and find the radius and centre of the circle.

I have a problem manipulating w to find the real part of w

Hi Punch,

Let \(z=x+yi\) where \(x,y\in\Re\).

\[w=\frac{z+1}{z-2i}=\frac{(x+1)+yi}{x+(y-2)i}\]

Now multiply both the numerator and the denominator by \(x-(y-2)i\). Hope you can continue. :)

 

[Punch]

Hi Punch,

Let \(z=x+yi\) where \(x,y\in\Re\).

\[w=\frac{z+1}{z-2i}=\frac{(x+1)+yi}{x+(y-2)i}\]

Now multiply both the numerator and the denominator by \(x-(y-2)i\). Hope you can continue. :)

I followed as you said but couldn't find how to extract the real part out...

 

[Sudharaka]

I followed as you said but couldn't find how to extract the real part out...

What did you get after multiplying by \(x-(y-2)i\) ?

 

[Punch]

What did you get after multiplying by \(x-(y-2)i\) ?

With regards to the real part, \frac{x^2+x+y^2+2y}{x^2+y^2-2y+4}=0

But I do not see how i can use this

 

[Sudharaka]

With regards to the real part, \[\frac{x^2+x+y^2+2y}{x^2+y^2-2y+4}=0\]

But I do not see how i can use this

\[Re(w)=\frac{x^2+x+y^2-2y}{x^2+y^2-4y+4}\]

As you may see, you have calculated the real part incorrectly. Check your calculation again.

Since, \(Re(w)=0\Rightarrow x^2+x+y^2-2y=0\)

Now can you try to transform this equation into the form \((x-a)^2+(y-b)^2=r^2\). The complex number \(z\) is on this circle. All you got to do is find \(a, b\) and \(c\).