Intersection of two loci

1. The complex number \(z\) is represented by the point \(P ( x , y )\) in an Argand diagram.

Two loci, \(L _ { 1 }\) and \(L _ { 2 }\), are given by: $$\begin{aligned} & L _ { 1 } : | z - 2 + \mathrm { i } | = | z + 2 - 3 \mathrm { i } | \\ & L _ { 2 } : | z - 2 + \mathrm { i } | = \sqrt { 10 } \end{aligned}$$

1. Find the coordinates of the points of intersection of these loci.
2. On the same Argand diagram, sketch the loci \(L _ { 1 }\) and \(L _ { 2 }\). Clearly label the coordinates of the points of intersection.
   

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 \(r\).

In an Argand diagram, the points \(A\) and \(B\) are represented by the complex numbers \(- 3 + 2 \mathrm { i }\) and \(5 - 4 \mathrm { i }\) respectively. The points \(A\) and \(B\) are the end points of a diameter of a circle \(C\).

1. Find the equation of \(C\), giving your answer in the form $$| z - a | = b \quad a \in \mathbb { C } , \quad b \in \mathbb { R }$$ The circle \(D\), with equation \(| z - 2 - 3 i | = 2\), intersects \(C\) at the points representing the complex numbers \(z _ { 1 }\) and \(z _ { 2 }\)
2. Find the complex numbers \(z _ { 1 }\) and \(z _ { 2 }\)

2

1. Sketch on one diagram:
   1. the locus of points satisfying \(| z - 4 + 2 \mathrm { i } | = 2\);
   2. the locus of points satisfying \(| z | = | z - 3 - 2 \mathrm { i } |\).
2. Shade on your sketch the region in which
   both $$| z - 4 + 2 i | \leqslant 2$$ and $$| z | \leqslant | z - 3 - 2 \mathrm { i } |$$

3 A circle \(C\) and a half-line \(L\) have equations $$| z - 2 \sqrt { 3 } - \mathrm { i } | = 4$$ and $$\arg ( z + i ) = \frac { \pi } { 6 }$$ respectively.

1. Show that:
   1. the circle \(C\) passes through the point where \(z = - \mathrm { i }\);
   2. the half-line \(L\) passes through the centre of \(C\).
2. On one Argand diagram, sketch \(C\) and \(L\).
3. Shade on your sketch the set of points satisfying both $$| z - 2 \sqrt { 3 } - \mathrm { i } | \leqslant 4$$ and $$0 \leqslant \arg ( z + i ) \leqslant \frac { \pi } { 6 }$$

4

1. On one Argand diagram, sketch the locus of points satisfying:
   1. \(| z - 3 + 2 \mathrm { i } | = 4\);
   2. \(\quad \arg ( z - 1 ) = - \frac { 1 } { 4 } \pi\).
2. Indicate on your sketch the set of points satisfying both $$| z - 3 + 2 i | \leqslant 4$$ and $$\arg ( z - 1 ) = - \frac { 1 } { 4 } \pi$$ (1 mark)

5 The sketch shows an Argand diagram. The points \(A\) and \(B\) represent the complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) respectively. The angle \(A O B = 90 ^ { \circ }\) and \(O A = O B\). \includegraphics[max width=\textwidth, alt={}, center]{847295e3-d806-43b1-8d25-688c5558bfe1-3_533_869_852_632}

1. Explain why \(z _ { 2 } = \mathrm { i } z _ { 1 }\).
2. On a single copy of the diagram, draw:
   1. the locus \(L _ { 1 }\) of points satisfying \(\left| z - z _ { 2 } \right| = \left| z - z _ { 1 } \right|\);
   2. the locus \(L _ { 2 }\) of points satisfying \(\arg \left( z - z _ { 2 } \right) = \arg z _ { 1 }\).
3. Find, in terms of \(z _ { 1 }\), the complex number representing the point of intersection of \(L _ { 1 }\) and \(L _ { 2 }\).

12

1. Sketch, on the Argand diagram below, the locus of points satisfying the equation $$| z - 2 \mathrm { i } | = 2$$
   \includegraphics[max width=\textwidth, alt={}]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-18_1219_1260_477_392}
   12
2. Sketch, also on the Argand diagram above, the locus of points satisfying the equation $$\arg z = \frac { \pi } { 3 }$$ [1 mark] 12
3. For the complex number \(w\) find the maximum value of \(| w |\) such that $$| w - 2 \mathrm { i } | \leq 2 \quad \text { and } \quad 0 \leq \arg w \leq \frac { \pi } { 3 }$$ $$y = \frac { 2 x + 7 } { 3 x + 5 }$$

17 The circle \(C\) represents the locus of points satisfying the equation $$| z - a \mathrm { i } | = b$$ where \(a\) and \(b\) are real constants. The circle \(C\) intersects the imaginary axis at 2 i and 8 i
The circle \(C\) is shown on the Argand diagram in Figure 2 \begin{figure}[h] \end{figure} 17

1. 1. Write down the value of \(a\) 17
      1. (ii) Write down the value of \(b\) 17
   2. The half-line \(L\) represents the locus of points satisfying the equation $$\arg ( z ) = \tan ^ { - 1 } ( k )$$ where \(k\) is a positive constant.
      The point \(P\) is the only point which lies on both \(C\) and \(L\), as shown in Figure 3 \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{47b12ae4-ca3f-472c-9d15-2ef17a2a4d87-25_766_770_685_699}
      \end{figure} 17
   3. 1. The point \(O\) represents the number \(0 + 0 \mathrm { i }\) Calculate the length \(O P\) 17
   4. (ii) Calculate the exact value of \(k\) 17
   5. (iii) Find the complex number represented by point \(P\) Give your answer in the form \(x + y i\) where \(x\) and \(y\) are real.

4 Two loci, \(C _ { 1 }\) and \(C _ { 2 }\), are defined by $$\begin{aligned} & C _ { 1 } = \left\{ z : | z | = \left| z - 4 d ^ { 2 } - 36 \right| \right\} \\ & C _ { 2 } = \left\{ z : \arg ( z - 12 d - 3 \mathrm { i } ) = \frac { 1 } { 4 } \pi \right\} \end{aligned}$$ where \(d\) is a real number.

1. Find, in terms of \(d\), the complex number which is represented on an Argand diagram by the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
   [0pt] [You may assume that \(C _ { 1 } \cap C _ { 2 } \neq \varnothing\).]
2. Explain why the solution found in part (a) is not valid when \(d = 3\).

3

1. On a single copy of an Argand diagram, sketch the loci defined by $$| z + 2 | = 3 \quad \text { and } \quad \arg ( z - \mathrm { i } ) = - \frac { 1 } { 4 } \pi$$
2. State the complex number \(z\) which corresponds to the point of intersection of these two loci.

The loci \(C_1\) and \(C_2\) are given by $$|z - 2i| = 2 \quad \text{and} \quad |z + 1| = |z + i|$$ respectively.

1. Sketch, on a single Argand diagram, the loci \(C_1\) and \(C_2\). [5]
2. Hence write down the complex numbers represented by the points of intersection of \(C_1\) and \(C_2\). [2]

The loci \(C_1\) and \(C_2\) are given by $$|z - 2\text{i}| = 2 \quad \text{and} \quad |z + 1| = |z + \text{i}|$$ respectively.

1. Sketch, on a single Argand diagram, the loci \(C_1\) and \(C_2\). [5]
2. Hence write down the complex numbers represented by the points of intersection of \(C_1\) and \(C_2\). [2]

Two loci, \(L_1\) and \(L_2\), in an Argand diagram are given by $$L_1 : |z + 6 - 5\text{i}| = 4\sqrt{2}$$ $$L_2 : \arg(z + \text{i}) = \frac{3\pi}{4}$$ The point \(P\) represents the complex number \(-2 + \text{i}\).

1. Verify that the point \(P\) is a point of intersection of \(L_1\) and \(L_2\). [2 marks]
2. Sketch \(L_1\) and \(L_2\) on one Argand diagram. [6 marks]
3. The point \(Q\) is also a point of intersection of \(L_1\) and \(L_2\). Find the complex number that is represented by \(Q\). [2 marks]

Two loci, \(C_1\) and \(C_2\), are defined by $$C_1 = \{z:|z| = |z - 4d^2 - 36|\}$$ $$C_2 = \left\{z:\arg(z - 12d - 3\text{i}) = \frac{1}{4}\pi\right\}$$ where \(d\) is a real number.

1. Find, in terms of \(d\), the complex number which is represented on an Argand diagram by the point of intersection of \(C_1\) and \(C_2\). [You may assume that \(C_1 \cap C_2 \neq \emptyset\).] [6]
2. Explain why the solution found in part (a) is not valid when \(d = 3\). [2]

Given that there are two distinct complex numbers \(z\) that satisfy $$\{z: |z - 3 - 5i| = 2r\} \cap \left\{z: \arg(z - 2) = \frac{3\pi}{4}\right\}$$ determine the exact range of values for the real constant \(r\). [7]

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 complex number \(z\) satisfies the equations $$|z^* - 1 - 2i| = |z - 3|$$ 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. [6 marks]

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. 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]