Geometric properties using complex numbers

9 The complex number 3 - i is denoted by \(u\). Its complex conjugate is denoted by \(u ^ { * }\).

1. On an Argand diagram with origin \(O\), show the points \(A , B\) and \(C\) representing the complex numbers \(u , u ^ { * }\) and \(u ^ { * } - u\) respectively. What type of quadrilateral is \(O A B C\) ?
2. Showing your working and without using a calculator, express \(\frac { u ^ { * } } { u }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
3. By considering the argument of \(\frac { u ^ { * } } { u }\), prove that $$\tan ^ { - 1 } \left( \frac { 3 } { 4 } \right) = 2 \tan ^ { - 1 } \left( \frac { 1 } { 3 } \right)$$

9 The complex number \(3 - \mathrm { i }\) is denoted by \(u\). Its complex conjugate is denoted by \(u ^ { * }\).

1. On an Argand diagram with origin \(O\), show the points \(A , B\) and \(C\) representing the complex numbers \(u , u ^ { * }\) and \(u ^ { * } - u\) respectively. What type of quadrilateral is \(O A B C\) ?
2. Showing your working and without using a calculator, express \(\frac { u ^ { * } } { u }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
3. By considering the argument of \(\frac { u ^ { * } } { u }\), prove that $$\tan ^ { - 1 } \left( \frac { 3 } { 4 } \right) = 2 \tan ^ { - 1 } \left( \frac { 1 } { 3 } \right) .$$ \includegraphics[max width=\textwidth, alt={}, center]{d4a7604c-9e2c-47ef-a496-8697bc88fdd4-18_360_758_260_689} The diagram shows the curve \(y = \frac { x ^ { 2 } } { 1 + x ^ { 3 } }\) for \(x \geqslant 0\), and its maximum point \(M\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = p\).
4. Find the exact value of the \(x\)-coordinate of \(M\).
5. Calculate the value of \(p\) for which the area of \(R\) is equal to 1 . Give your answer correct to 3 significant figures.

7 In an Argand diagram the points representing the numbers \(2 + 3 \mathrm { i }\) and \(1 - \mathrm { i }\) are two adjacent vertices of a square, \(S\).

1. Find the area of \(S\).
2. Find all the possible pairs of numbers represented by the other two vertices of \(S\).

7. $$f ( z ) = z ^ { 3 } + z ^ { 2 } + p z + q$$ where \(p\) and \(q\) are real constants.
The equation \(f ( z ) = 0\) has roots \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) When plotted on an Argand diagram, the points representing \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) form the vertices of a triangle of area 35 Given that \(z _ { 1 } = 3\), find the values of \(p\) and \(q\).

7. $$f ( z ) = z ^ { 4 } + a z ^ { 3 } + b z ^ { 2 } + c z + d$$ where \(a\), \(b\), \(c\) and \(d\) are real constants.
The equation \(\mathrm { f } ( \mathrm { z } ) = 0\) has complex roots \(\mathrm { z } _ { 1 } , \mathrm { z } _ { 2 } , \mathrm { z } _ { 3 }\) and \(\mathrm { z } _ { 4 }\) When plotted on an Argand diagram, the points representing \(z _ { 1 } , z _ { 2 } , z _ { 3 }\) and \(z _ { 4 }\) form the vertices of a square, with one vertex in each quadrant.
Given that \(z _ { 1 } = 2 + 3 i\), determine the values of \(a , b , c\) and \(d\).

Given that \(z_1 = 3 + 2i\) and \(z_2 = \frac{12 - 5i}{z_1}\).

1. Find \(z_2\) in the form \(a + ib\), where \(a\) and \(b\) are real. [2]
2. Show, on an Argand diagram, the point \(P\) representing \(z_1\) and the point \(Q\) representing \(z_2\). [2]
3. Given that \(O\) is the origin, show that \(\angle POQ = \frac{\pi}{2}\). [2]

The circle passing through the points \(O\), \(P\) and \(Q\) has centre \(C\). Find

1. the complex number represented by \(C\), [2]
2. the exact value of the radius of the circle. [2]

Given that \(z = 3 + 4i\) and \(w = -1 + 7i\).

1. find \(|w|\). [1]

The complex numbers \(z\) and \(w\) are represented by the points \(A\) and \(B\) on an Argand diagram.

1. Show points \(A\) and \(B\) on an Argand diagram. [1]
2. Prove that \(\triangle OAB\) is an isosceles right-angled triangle. [5]
3. Find the exact value of \(\arg \left( \frac{z}{w} \right)\). [3]

Given that \(z = 3 - 3i\) express, in the form \(a + ib\), where \(a\) and \(b\) are real numbers,

1. \(z^2\), [2]
2. \(\frac{1}{z}\), [2]
3. Find the exact value of each of \(|z|\), \(|z^2|\) and \(\left|\frac{1}{z}\right|\). [2]

The complex numbers \(z\), \(z^2\) and \(\frac{1}{z}\) are represented by the points \(A\), \(B\) and \(C\) respectively on an Argand diagram. The real number 1 is represented by the point \(D\), and \(O\) is the origin.

1. Show the points \(A\), \(B\), \(C\) and \(D\) on an Argand diagram. [2]
2. Prove that \(\triangle OAB\) is similar to \(\triangle OCD\). [3]

In an Argand diagram the points representing the numbers \(2 + 3i\) and \(1 - i\) are two adjacent vertices of a square, \(S\).

1. Find the area of \(S\). [3]
2. Find all the possible pairs of numbers represented by the other two vertices of \(S\). [4]