91Ó°ÊÓ

Complex numbers can be represented in the form of \( z = x + yi \), where \( x \) is the real part and \( yi \) is the imaginary part. The modulus of a complex number is a measure of its absolute size or magnitude on the complex plane. Imagine it as the distance from the origin to the point \( z \) in the complex plane. This can be calculated using the formula:\[|z| = \sqrt{x^2 + y^2}\]For example, if we have \( z = 3 + 4i \), the modulus would be \( |z| = \sqrt{3^2 + 4^2} = 5 \). In the given problem, the equation \( |z - i| = 5 \) shows the distance from \( z \) to the point \( i \) (which is the point \( (0, 1) \) in the complex plane) is 5. By applying the modulus condition, you can see how it confines the location of \( z \) to lie on a circle of radius 5 centered at \( i \) in the complex plane.
To simplify this equation, you would express it as \( \sqrt{(x-0)^2 + (y-1)^2} = 5 \) and then square both sides to eventually form the quadratic equation.

The argument of a complex number provides the angle formed with the positive real axis when the complex number is represented in polar coordinates. For a complex number \( z = x + yi \), the argument, \( \arg(z) \), is calculated with:\[\tan^{-1}\left(\frac{y}{x}\right)\]This value is essentially the slope of the line connecting \( z \) to the origin. In trigonometric terms, it’s often helpful because it shows directionality. In our original exercise, the condition \( \arg(z) = \pi/4 \) tells us that the angle between the line connecting \( z \) to the origin and the positive real axis is \( 45^\circ \).
This means \( \frac{y}{x} = 1 \), implying \( y = x \). This condition tells us more about the alignment of points along a line in the complex plane, directly influencing how we solve for \( z \). By substituting \( y = x \) into our earlier equation concerning the modulus, we simplify the process of finding the solution.

Quadratic equations are polynomial equations of the form \( ax^2 + bx + c = 0 \). Solving these equations often involves finding values for \( x \) that make the equation true. In our exercise, after substituting \( y = x \) into the modulus equation, we derive the quadratic equation \( x^2 - x = 12 \), or equivalently \( x^2 - x - 12 = 0 \).
To solve this, we factor it into \( (x - 4)(x + 3) = 0 \). This gives the solutions \( x = 4 \) and \( x = -3 \). These solutions represent the x-values where the original conditions of modulus and argument are satisfied. Once we have \( x \), finding \( y \) becomes straightforward since \( y = x \). Thus, the solutions for \( z \) are \( z = 4 + 4i \) and \( z = -3 - 3i \). These are the complex numbers which solve the simultaneous equations given their alignment with the problem's conditions.