Cartesian equation from argument condition

9. The complex number \(z\) is represented by the point \(P\) in an Argand diagram. Given that \(\arg \left( \frac { z - 5 } { z - 2 } \right) = \frac { \pi } { 4 }\)

1. sketch the locus of \(P\) as \(z\) varies,
2. find the exact maximum value of \(| z |\).
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1. A complex number \(z\) is represented by the point \(P\) on an Argand diagram.

Given that \(\arg \left( \frac { z - 6 i } { z - 3 i } \right) = \frac { \pi } { 3 }\)

1. sketch the locus of \(P\) as \(z\) varies,
2. find the exact maximum possible value of \(| z |\)

1. A complex number \(z\) is represented by the point \(P\) on an Argand diagram.

Given that $$\arg \left( \frac { z - 4 - i } { z - 2 - 7 i } \right) = \frac { \pi } { 2 }$$

1. sketch the locus of \(P\) as \(z\) varies,
2. determine the exact maximum possible value of \(| z |\)

1. The locus of points \(z = x + \mathrm { i } y\) that satisfy

$$\arg \left( \frac { z - 8 - 5 i } { z - 2 - 5 i } \right) = \frac { \pi } { 3 }$$ is an arc of a circle \(C\).

1. On an Argand diagram sketch the locus of \(z\).
2. Explain why the centre of \(C\) has \(x\) coordinate 5
3. Determine the radius of \(C\).
4. Determine the \(y\) coordinate of the centre of \(C\).

9. \begin{figure}[h] \end{figure} Figure 1 shows a locus in the complex plane.
The locus is an arc of a circle from the point represented by \(z _ { 1 } = 3 + 2 i\) to the point represented by \(z _ { 2 } = a + 4 \mathrm { i }\), where \(a\) is a constant, \(a \neq 1\) Given that

• the point \(z _ { 3 } = 1 + 4 \mathrm { i }\) also lies on the locus
• the centre of the circle has real part equal to - 1
  1. determine the value of \(a\).
  2. Hence determine a complex equation for the locus, giving any angles in the equation as positive values.