Optimization on loci

7 The complex number \(2 + 2 \mathrm { i }\) is denoted by \(u\).

1. Find the modulus and argument of \(u\).
2. Sketch an Argand diagram showing the points representing the complex numbers 1, i and \(u\). Shade the region whose points represent the complex numbers \(z\) which satisfy both the inequalities \(| z - 1 | \leqslant | z - \mathrm { i } |\) and \(| z - u | \leqslant 1\).
3. Using your diagram, calculate the value of \(| z |\) for the point in this region for which \(\arg z\) is least.

8 The complex number \(u\) is defined by \(u = \frac { 6 - 3 \mathrm { i } } { 1 + 2 \mathrm { i } }\).

1. Showing all your working, find the modulus of \(u\) and show that the argument of \(u\) is \(- \frac { 1 } { 2 } \pi\).
2. For complex numbers \(z\) satisfying \(\arg ( z - u ) = \frac { 1 } { 4 } \pi\), find the least possible value of \(| z |\).
3. For complex numbers \(z\) satisfying \(| z - ( 1 + \mathrm { i } ) u | = 1\), find the greatest possible value of \(| z |\).

10

1. Showing all necessary working, solve the equation \(\mathrm { i } z ^ { 2 } + 2 z - 3 \mathrm { i } = 0\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
2. 1. On a sketch of an Argand diagram, show the locus representing complex numbers satisfying the equation \(| z | = | z - 4 - 3 \mathrm { i } |\).
   2. Find the complex number represented by the point on the locus where \(| z |\) is least. Find the modulus and argument of this complex number, giving the argument correct to 2 decimal places.

7 The complex number \(u\) is given by \(u = \frac { 7 + 4 \mathrm { i } } { 3 - 2 \mathrm { i } }\).

1. Express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. Sketch an Argand diagram showing the point representing the complex number \(u\). Show on the same diagram the locus of the complex number \(z\) such that \(| z - u | = 2\).
3. Find the greatest value of \(\arg z\) for points on this locus.

9 The complex number \(u\) is given by $$u = \frac { 3 + \mathrm { i } } { 2 - \mathrm { i } }$$

1. Express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. Find the modulus and argument of \(u\).
3. Sketch an Argand diagram showing the point representing the complex number \(u\). Show on the same diagram the locus of the point representing the complex number \(z\) such that \(| z - u | = 1\).
4. Using your diagram, calculate the least value of \(| z |\) for points on this locus.

3

1. On an Argand diagram, sketch the locus of points representing complex numbers \(z\) satisfying \(| z + 3 - 2 \mathrm { i } | = 2\).
2. Find the least value of \(| z |\) for points on this locus, giving your answer in an exact form.

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 \(| z |\).
   VILM SIHI NITIIIUMI ON OC
   VILV SIHI NI III HM ION OC
   VALV SIHI NI JIIIM ION OO

8. A complex number \(z\) satisfies the equation $$| z - 5 - 12 i | = 3$$

1. Describe in geometrical terms with the aid of a sketch, the locus of the point which represents \(z\) in the A rgand diagram. For points on this locus, find
2. the maximum and minimum values for \(| z |\),
3. the maximum and minimum values for arg \(z\), giving your answers in radians to 2 decimal places.

6 In an Argand diagram, the variable point \(P\) represents the complex number \(z = x + \mathrm { i } y\), and the fixed point \(A\) represents \(a = 4 - 3 \mathrm { i }\).

1. Sketch an Argand diagram showing the position of \(A\), and find \(| a |\) and \(\arg a\).
2. Given that \(| z - a | = | a |\), sketch the locus of \(P\) on your Argand diagram.
3. Hence write down the non-zero value of \(z\) corresponding to a point on the locus for which
   (a) the real part of \(z\) is zero,
   (b) \(\quad \arg z = \arg a\).

8

1. Sketch on an Argand diagram the locus, \(C\), of points for which \(| z - 4 | = 3\).
2. By drawing appropriate lines through the origin, indicate on your Argand diagram the point A on the locus \(C\) where \(\arg z\) has its maximum value. Indicate also the point B on the locus \(C\) where \(\arg z\) has its minimum value.
3. Given that \(\arg z = \alpha\) at A and \(\arg z = \beta\) at B , indicate on your Argand diagram the set of points for which \(\beta \leqslant \arg z \leqslant \alpha\) and \(| z - 4 | \geqslant 3\).
4. Calculate the value of \(\alpha\) and the value of \(\beta\).

8

1. Indicate on an Argand diagram the set of points \(z\) for which \(| z - ( - 8 + 15 \mathrm { j } ) | < 10\).
2. Using the diagram, show that \(7 < | z | < 27\).
3. Mark on your Argand diagram the point, \(P\), at which \(| z - ( - 8 + 15 \mathrm { j } ) | = 10\) and \(\arg z\) takes its maximum value. Find the modulus and argument of \(z\) at \(P\).

8 You are given the complex number \(w = 2 + 2 \sqrt { 3 } \mathrm { j }\).

1. Express \(w\) in modulus-argument form.
2. Indicate on an Argand diagram the set of points, \(z\), which satisfy both of the following inequalities. $$- \frac { \pi } { 2 } \leqslant \arg z \leqslant \frac { \pi } { 3 } \text { and } | z | \leqslant 4$$ Mark \(w\) on your Argand diagram and find the greatest value of \(| z - w |\).

14

1. Sketch, on the Argand diagram below, the locus of points satisfying the equation $$| z - 3 | = 2$$
   \includegraphics[max width=\textwidth, alt={}]{1d017497-11b1-4096-b83a-63314188307e-16_1216_1251_486_392}
   14
2. There is a unique complex number \(w\) that satisfies both $$| w - 3 | = 2 \quad \text { and } \quad \arg ( w + 1 ) = \alpha$$ where \(\alpha\) is a constant such that \(0 < \alpha < \pi\) 14
3. 1. Find the value of \(\alpha\).
      14
4. (ii) Express \(w\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\).
   Give each of \(r\) and \(\theta\) to two significant figures.
   1. (a) Show that
   $$\frac { 1 } { r + 2 } - \frac { 1 } { r + 3 } = \frac { 1 } { ( r + 2 ) ( r + 3 ) }$$

1

1. Sketch on an Argand diagram the locus of points satisfying the equation $$| z - 4 + 3 \mathrm { i } | = 5$$
2. 1. Indicate on your diagram the point \(P\) representing \(z _ { 1 }\), where both $$\left| z _ { 1 } - 4 + 3 \mathrm { i } \right| = 5 \quad \text { and } \quad \arg z _ { 1 } = 0$$
   2. Find the value of \(\left| z _ { 1 } \right|\).

2

1. Draw on an Argand diagram the locus \(L\) of points satisfying the equation \(\arg z = \frac { \pi } { 6 }\).
   (1 mark)
2. 1. A circle \(C\), of radius 6, has its centre lying on \(L\) and touches the line \(\operatorname { Re } ( z ) = 0\). Draw \(C\) on your Argand diagram from part (a).
   2. Find the equation of \(C\), giving your answer in the form \(\left| z - z _ { 0 } \right| = k\).
   3. The complex number \(z _ { 1 }\) lies on \(C\) and is such that \(\arg z _ { 1 }\) has its least possible value. Find \(\arg z _ { 1 }\), giving your answer in the form \(p \pi\), where \(- 1 < p \leqslant 1\).

1

1. Sketch on an Argand diagram the locus of points satisfying the equation $$| z - 6 \mathrm { i } | = 3$$
2. It is given that \(z\) satisfies the equation \(| z - 6 \mathrm { i } | = 3\).
   1. Write down the greatest possible value of \(| z |\).
   2. Find the greatest possible value of \(\arg z\), giving your answer in the form \(p \pi\), where \(- 1 < p \leqslant 1\).

1. Given that there are two distinct complex numbers \(z\) that satisfy

$$\{ z : | z - 3 - 5 i | = 2 r \} \cap \quad z : \arg ( z - 2 ) = \frac { 3 \pi } { 4 }$$ determine the exact range of values for the real constant \(r\).

1. (a) Shade on an Argand diagram the set of points

$$\{ z \in \mathbb { C } : | z - 4 i | \leqslant 3 \} \cap \left\{ z \in \mathbb { C } : - \frac { \pi } { 2 } < \arg ( z + 3 - 4 i ) \leqslant \frac { \pi } { 4 } \right\}$$ The complex number \(w\) satisfies $$| w - 4 \mathrm { i } | = 3$$ (b) Find the maximum value of \(\arg w\) in the interval \(( - \pi , \pi ]\). Give your answer in radians correct to 2 decimal places.

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 \(| z |\)

5. \begin{figure}[h] \end{figure} Figure 1 shows an Argand diagram.
The set of points, \(A\), that lies within the shaded region, including its boundaries, is defined by $$A = \{ z : p \leqslant \arg ( z ) \leqslant q \} \cap \{ z : | z | \leqslant r \}$$ where \(p , q\) and \(r\) are positive constants.

1. Write down the values of \(p , q\) and \(r\). Given that \(w = - 2 \sqrt { 3 } + 2 \mathrm { i }\) and \(\mathrm { z } \in A\),
2. find the maximum value of \(| w - z | ^ { 2 }\) giving your answer in an exact simplified form.

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. A circle \(C\) in the complex plane has equation

$$| z - ( - 3 + 3 i ) | = \alpha | z - ( 1 + 3 i ) |$$ where \(\alpha\) is a real constant with \(\alpha > 1\) Given that the imaginary axis is a tangent to \(C\)

1. sketch, on an Argand diagram, the circle \(C\)
2. explain why the value of \(\alpha\) is 3 The circle \(C\) is contained in the region $$R = \left\{ z \in \mathbb { C } : \beta \leqslant \arg z \leqslant \frac { \pi } { 2 } \right\}$$
3. Determine the maximum value of \(\beta\) Give your answer in radians to 3 significant figures.

1. (i) Given that

$$z _ { 1 } = 6 \mathrm { e } ^ { \frac { \pi } { 3 } \mathrm { i } } \text { and } z _ { 2 } = 6 \sqrt { 3 } \mathrm { e } ^ { \frac { 5 \pi } { 6 } \mathrm { i } }$$ show that $$z _ { 1 } + z _ { 2 } = 12 \mathrm { e } ^ { \frac { 2 \pi } { 3 } \mathrm { i } }$$ (ii) Given that $$\arg ( z - 5 ) = \frac { 2 \pi } { 3 }$$ determine the least value of \(| z |\) as \(z\) varies.

5 The complex number \(z\) satisfies the relation $$| z + 4 - 4 i | = 4$$

1. Sketch, on an Argand diagram, the locus of \(z\).
2. Show that the greatest value of \(| z |\) is \(4 ( \sqrt { 2 } + 1 )\).
3. Find the value of \(z\) for which $$\arg ( z + 4 - 4 \mathrm { i } ) = \frac { 1 } { 6 } \pi$$ Give your answer in the form \(a + \mathrm { i } b\).