Optimization on loci

2

1. Indicate on an Argand diagram the region for which \(| z - 4 \mathrm { i } | \leqslant 2\).
2. The complex number \(z\) satisfies \(| z - 4 \mathrm { i } | \leqslant 2\). Find the range of possible values of \(\arg z\).

6

1. Two points, \(A\) and \(B\), on an Argand diagram are represented by the complex numbers \(2 + 3 \mathrm { i }\) and \(- 4 - 5 \mathrm { i }\) respectively. Given that the points \(A\) and \(B\) are at the ends of a diameter of a circle \(C _ { 1 }\), express the equation of \(C _ { 1 }\) in the form \(\left| z - z _ { 0 } \right| = k\).
2. A second circle, \(C _ { 2 }\), is represented on the Argand diagram by the equation \(| z - 5 + 4 \mathrm { i } | = 4\). Sketch on one Argand diagram both \(C _ { 1 }\) and \(C _ { 2 }\).
3. The points representing the complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) lie on \(C _ { 1 }\) and \(C _ { 2 }\) respectively and are such that \(\left| z _ { 1 } - z _ { 2 } \right|\) has its maximum value. Find this maximum value, giving your answer in the form \(a + b \sqrt { 5 }\).

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 $$\arg ( z + 4 - 4 i ) = \frac { 1 } { 6 } \pi$$ Give your answer in the form \(a + \mathrm { i } b\).

18 The locus of points \(L _ { 1 }\) satisfies the equation \(| z | = 2\) The locus of points \(L _ { 2 }\) satisfies the equation \(\arg ( z + 4 ) = \frac { \pi } { 4 }\) 18

1. Sketch \(L _ { 1 }\) on the Argand diagram below.
   [0pt] [1 mark] \includegraphics[max width=\textwidth, alt={}, center]{86aa9e6f-261c-40d4-8271-a0dc560d8a72-26_1152_1195_644_427} 18
2. Sketch \(L _ { 2 }\) on the Argand diagram above.
   [0pt] [1 mark] 18
3. The complex number \(a + \mathrm { i } b\), where \(a\) and \(b\) are real, lies on \(L _ { 1 }\) The complex number \(c + \mathrm { i } d\), where \(c\) and \(d\) are real, lies on \(L _ { 2 }\) Calculate the least possible value of the expression $$( c - a ) ^ { 2 } + ( d - b ) ^ { 2 }$$ \includegraphics[max width=\textwidth, alt={}, center]{86aa9e6f-261c-40d4-8271-a0dc560d8a72-28_2492_1721_217_150}

8 The complex number \(z\) satisfies the equations $$\left| z ^ { * } - 1 - 2 i \right| = | z - 3 |$$ and $$| z - a | = 3$$ where \(a\) is real.
Show that \(a\) must lie in the interval \([ 1 - s \sqrt { t } , 1 + s \sqrt { t } ]\), where \(s\) and \(t\) are prime numbers.
[0pt] [6 marks]

10

1. Sketch \(R\) on the Argand diagram below.
   [0pt] [3 marks] \includegraphics[max width=\textwidth, alt={}, center]{bc1b33a7-800b-4359-b7ba-6460f17984e5-10_1205_1200_520_422} 10
2. Find the maximum value of \(| z |\) in the region \(R\), giving your answer in exact form.

17 The Argand diagram below shows a circle \(C\) \includegraphics[max width=\textwidth, alt={}, center]{99b03f18-6dd6-437d-8b01-009ca7ab49ea-22_1063_926_317_541} 17

1. Write down the equation of the locus of \(C\) in the form $$| z - w | = a$$ where \(w\) is a complex number whose real and imaginary parts are integers, and \(a\) is an integer.
   17
2. It is given that \(z _ { 1 }\) is a complex number representing a point on \(C\). Of all the complex numbers which represent points on \(C , z _ { 1 }\) has the least argument. 17
3. 1. Find \(\left| z _ { 1 } \right|\) Give your answer in an exact form.
      17
4. (ii) Show that \(\arg z _ { 1 } = \arcsin \left( \frac { 6 \sqrt { 3 } - 2 } { 13 } \right)\)
   \includegraphics[max width=\textwidth, alt={}]{99b03f18-6dd6-437d-8b01-009ca7ab49ea-25_2486_1744_178_132}