4.02o Loci in Argand diagram: circles, half-lines

In an Argand diagram the loci \(C_1\) and \(C_2\) are given by $$|z| = 2 \quad \text{and} \quad \arg z = \frac{1}{4}\pi$$ respectively.

1. Sketch, on a single Argand diagram, the loci \(C_1\) and \(C_2\). [5]
2. Hence find, in the form \(x + iy\), the complex number representing the point of intersection of \(C_1\) and \(C_2\). [2]

The point \(P\) represents a complex number \(z\) on an Argand diagram such that $$|z - 6i| = 2|z - 3|.$$ Show that, as \(z\) varies, the locus of \(P\) is a circle, stating the radius and the coordinates of the centre of this circle. [6 marks]

\includegraphics{figure_11} Figure 1 shows an Argand diagram. The set \(P\) of points that lie within the shaded region including its boundaries, is defined by $$P = \{z \in \mathbb{C} : a \leq |z + b + ci| \leq d\}$$ where \(a\), \(b\), \(c\) and \(d\) are integers.

1. Write down the values of \(a\), \(b\), \(c\) and \(d\). [3]

The set \(Q\) is defined by $$Q = \{z \in \mathbb{C} : a \leq |z + b + ci| \leq d\} \cap \{z \in \mathbb{C} : |z - i| \leq |z - 3i|\}$$

1. Determine the exact area of the region defined by \(Q\), giving your answer in simplest form. [7]

Find, in exact form, the area of the region on an Argand diagram which represents the locus of points for which \(|z - 5 - 2i| \leq \sqrt{32}\) and Re (z) \(\geq\) 9. [6]

1. Shade on an Argand diagram the set of points $$\left\{z \in \mathbb{C} : |z - 4i| \leqslant 3\right\} \cap \left\{z \in \mathbb{C} : -\frac{\pi}{2} < \arg(z + 3 - 4i) \leqslant \frac{\pi}{4}\right\}$$ [5]

The complex number \(w\) satisfies \(|w - 4i| = 3\).

1. Find the maximum value of \(\arg w\) in the interval \((-\pi, \pi]\). Give your answer in radians correct to 2 decimal places. [2]

1. Sketch, on the Argand diagram below, the locus of points satisfying the equation \(|z - 3| = 2\) [1]

\includegraphics{figure_9}

1. There is a unique complex number \(w\) that satisfies both \(|w - 3| = 2\) and \(\arg(w + 1) = \alpha\) where \(\alpha\) is a constant such that \(0 < \alpha < \pi\).
   1. Find the value of \(\alpha\). [2]
   2. Express \(w\) in the form \(r(\cos \theta + i \sin \theta)\). Give each of \(r\) and \(\theta\) to two significant figures. [4]

The point \(P\) represents a complex number \(z\) on an Argand diagram such that $$|z - 6i| = 2|z - 3|$$

1. Show that, as \(z\) varies, the locus of \(P\) is a circle, stating the radius and the coordinates of the centre of this circle. [6]

The point \(Q\) represents a complex number \(z\) on an Argand diagram such that $$\arg(z - 6) = -\frac{3\pi}{4}$$

1. Sketch, on the same Argand diagram, the locus of \(P\) and the locus of \(Q\) as \(z\) varies. [4]
2. Find the complex number for which both \(|z - 6i| = 2|z - 3|\) and \(\arg(z - 6) = -\frac{3\pi}{4}\). [4]

1. On the Argand diagram below, sketch the locus, \(L\), of points satisfying the equation $$\arg(z + i) = \frac{\pi}{6}$$ [2 marks]

\includegraphics{figure_5}

1. \(z_1\) is a point on \(L\) such that \(|z_1|\) is a minimum. Find the exact value of \(z_1\) in the form \(a + bi\) [4 marks]

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

1. Express \(w\) in modulus-argument form. [3]
2. Indicate on an Argand diagram the set of points, \(z\), which satisfy both of the following inequalities. $$-\frac{\pi}{2} \leq \arg z \leq \frac{\pi}{3} \text{ and } |z| \leq 4$$ Mark \(w\) on your Argand diagram and find the greatest value of \(|z - w|\). [9]

Find, in exact form, the area of the region on an Argand diagram which represents the locus of points for which \(|z - 5 - 2i| \leq \sqrt{32}\) and \(\text{Re}(z) \geq 9\). [6]

A locus \(C_1\) is defined by \(C_1 = \{z : |z + i| \leq |z - 2i|\}\).

1. Indicate by shading on the Argand diagram below the region representing \(C_1\). [2] \includegraphics{figure_8}
2. Find the cartesian equation of the boundary line of the region representing \(C_1\), giving your answer in the form \(ax + by + c = 0\). [2]

1. Sketch, on the Argand diagram below, the locus of points satisfying the equation $$|z - 3| = 2$$ [1 mark] \includegraphics{figure_8}

1. 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 \(0 < \alpha < \pi\)
   1. [(b) (i)] Find the value of \(\alpha\). [2 marks]
   2. [(b) (ii)] Express \(w\) in the form \(r(\cos \theta + i \sin \theta)\). [4 marks]

\(C\) is the locus of numbers, \(z\), for which \(\ln\left(\frac{z + 7i}{z - 24}\right) = \frac{1}{4}\). By writing \(z = x + iy\) give a complete description of the shape of \(C\) on an Argand diagram. [7]

The loci \(C_1\) and \(C_2\) are given by \(|z - (3 + 2i)| = 2\) and \(\arg(z - (3 + 2i)) = \frac{5\pi}{6}\) respectively.

1. Sketch \(C_1\) and \(C_2\) on a single Argand diagram. [4]
2. Find, in surd form, the number represented by the point of intersection of \(C_1\) and \(C_2\). [3]
3. Indicate, by shading, the region of the Argand diagram for which $$|z - (3 + 2i)| \leq 2 \text{ and } \frac{5\pi}{6} \leq \arg(z - (3 + 2i)) \leq \pi.$$ [2]

C is the locus of numbers, \(z\), for which \(\text{Im}\left(\frac{z + 7i}{z - 24}\right) = \frac{1}{4}\). By writing \(z = x + iy\) give a complete description of the shape of C on an Argand diagram. [7]

The region \(R\) of an Argand diagram is defined by the inequalities $$0 \leqslant \arg(z + 4\mathrm{i}) \leqslant \frac{1}{4}\pi \quad \text{and} \quad |z| \leqslant 4.$$ Draw a clearly labelled diagram to illustrate \(R\). [4]