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Curves and Regions - Questions

Question 1
52083

Define the region \(|z+i| \le |z-2|\)

Question 2
126948

Given that \(\omega=\dfrac{z+1}{z-i}\) and \(\omega\) is purely imaginary, find the locus of \(w\).

\(x(x+1)+y(y-1)=0\)

Question 3
126949

Sketch the region defined by \(|z+1| \leqslant 1\) and \(\operatorname{Im}(z) \geqslant 0\)

Refer to worked solution

Question 4
12863

\(z = x + iy\) is such that \(\dfrac{{z + i}}{{z + 1}}\) is purely imaginary. The equation of the locus of the point P representing \(z\) is?

\({\left( {x + \dfrac{1}{2}} \right)^2} + {\left( {y + \dfrac{1}{2}} \right)^2} = \dfrac{1}{2}\)

Question 5
12864

Given that \((z - 2)(\bar z - 2) = 4\) represents a circle, then the radius and coordinates of the centre are?

radius 2, centre (2,0)

Question 6
12826

\(\left| {z - i} \right|{\rm{ > }}\left| {z + 1} \right|{\rm{ }}\) represents the region :

\(x + y < 0\)

Question 7
12827

Given that \({\rm{ }}\omega = \dfrac{{z - 2}}{z}{\rm{ }}\) and \({\rm{ }}\left| z \right| = 1{\rm{ }}\) describes a circle, then the centre and radius is?

\({\rm{Centre }}\left( {1,0} \right),{\rm{ radius 2}}\)

Question 8
12828

Given that \(\omega = \dfrac{{z + 1}}{z}{\rm{ }}\) and \({\rm{ }}\omega {\rm{ }}\) is purely real then the locus of  \({\rm{ }}\omega {\rm{ }}\)  is?

\(y = 0,\,\,\,(0,0){\rm{ excluded}}\)

Question 9
12829

Given that \(\omega = \dfrac{{z - i}}{{z - 2}}{\rm{ }}\) and \({\rm{ }}\omega {\rm{ }}\) is purely imaginary then the locus of \({\rm{ }}\omega {\rm{ }}\) is?

\({\left( {x - 1} \right)^2} + {\left( {y - \dfrac{1}{2}} \right)^2} = \dfrac{5}{4}\)

Question 10
12273

\(2\left| z \right| = z + \mathop z\limits^ - + 2\) represents the cartesian equation:

A.

\({y^2} = 2x - 1\)

B.

\({y^2} = 2x + 1\)

C.

\(y = 2{x^2} + 1\)

D.

\(y = 2{x^2} - 1\)

\({y^2} = 2x + 1\)

Question 11
21636

Given that \(w = \dfrac{{z + 3}}{z}\) and \(|z| = 1\) describes a circle, find the centre and radius of the circle. 

Centre \((1,0)\), radius \(1\)

Question 12
26297

What region on the number plane is represented by \(|z + i| < |z - 1|\)?

\(x+y<0\)