Circle of Apollonius locus

7 The complex number \(z\) is defined by \(z = a + \mathrm { i } b\), where \(a\) and \(b\) are real. The complex conjugate of \(z\) is denoted by \(z ^ { * }\).

1. Show that \(| z | ^ { 2 } = z z ^ { * }\) and that \(( z - k \mathrm { i } ) ^ { * } = z ^ { * } + k \mathrm { i }\), where \(k\) is real. In an Argand diagram a set of points representing complex numbers \(z\) is defined by the equation \(| z - 10 \mathrm { i } | = 2 | z - 4 \mathrm { i } |\).
2. Show, by squaring both sides, that $$z z ^ { * } - 2 \mathrm { i } z ^ { * } + 2 \mathrm { i } z - 12 = 0$$ Hence show that \(| z - 2 i | = 4\).
3. Describe the set of points geometrically.

6. The complex number \(z\) is represented by the point \(P ( x , y )\) in an Argand diagram. Given that $$| z - 3 + \mathrm { i } | = 2 | z - 5 - 2 \mathrm { i } |$$ show that the locus of \(P\) is a circle and write down the coordinates of its centre.

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. A curve \(C\) is described by the equation

$$| z - 9 + 12 i | = 2 | z |$$

1. Show that \(C\) is a circle, and find its centre and radius.
2. Sketch \(C\) on an Argand diagram. Given that \(w\) lies on \(C\),
3. find the largest value of \(a\) and the smallest value of \(b\) that must satisfy $$a \leqslant \operatorname { Re } ( w ) \leqslant b$$

1. A complex number \(z = x + \mathrm { i } y\) is represented by the point \(P\) in an Argand diagram.

Given that $$| z - 3 | = 4 | z + 1 |$$

1. show that the locus of \(P\) has equation $$15 x ^ { 2 } + 15 y ^ { 2 } + 38 x + 7 = 0$$
2. Hence find the maximum value of \(| z |\)

1. The locus of points \(z\) satisfies

$$| z + a \mathrm { i } | = 3 | z - a |$$ where \(a\) is an integer.
The locus is a circle with its centre in the third quadrant and radius \(\frac { 3 } { 2 } \sqrt { 2 }\) Determine

1. the value of \(a\),
2. the coordinates of the centre of the circle.

1. A curve has equation

$$| z + 6 | = 2 | z - 6 | \quad z \in \mathbb { C }$$

1. Show that the curve is a circle with equation \(x ^ { 2 } + y ^ { 2 } - 20 x + 36 = 0\)
2. Sketch the curve on an Argand diagram. The line \(l\) has equation \(a z ^ { * } + a ^ { * } z = 0\), where \(a \in \mathbb { C }\) and \(z \in \mathbb { C }\) Given that the line \(l\) is a tangent to the curve and that \(\arg a = \theta\)
3. find the possible values of \(\tan \theta\)

The complex number \(z\) is represented by the point \(P(x, y)\) in an Argand diagram and $$|z - 3| = 2|z + i|.$$ Show that the locus of \(P\) is a circle and determine its radius and the coordinates of its centre. [9]

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]

\(C\) 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]

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]