Question

The equation of the set of complex number \(z=x+\) iy, So that \(\left|z-z_{1}\right|=5\), where \(z_{1}=1+2 i\) (a) \(x^{2}+y^{2}-2 x-4 y-20=0\) (b) \(x^{2}+y^{2}+2 x-4 y-20=0\) (c) \(x^{2}+y^{2}-2 x+4 y-20=0\) (d) \(x^{2}+y^{2}+2 x+4 y+20=0\)

Short answer

The short answer to the problem is: (a) \( x^2 + y^2 - 2x - 4y - 20 = 0 \)

Step-by-step solution

1. Write the equation for |z - z1|=

\( \sqrt{(x - 1)^2 + (y - 2)^2} = 5\) Now, we can eliminate the square root by squaring both sides of the equation:

2. Square both sides of the equation

\( (x - 1)^2 + (y - 2)^2 = 25 \) Now, expanding the squares and collecting like terms:

3. Expand the squares and simplify the equation

\( x^2 - 2x + 1 + y^2 - 4y + 4 = 25 \)

4. Rearrange the equation

\( x^2 - 2x + y^2 - 4y = 20 \) As we can see, the correct equation of the set of complex numbers z is: \( x^2 - 2x + y^2 - 4y - 20 = 0 \) By comparing our result with the given options, we can conclude that the correct answer is: (a) \( x^2 + y^2 - 2x - 4y - 20 = 0 \)

Key concepts

Equations of Circles

When dealing with complex numbers, equations of circles arise frequently. A circle is the set of all points in a plane that are equidistant from a given point, known as the center. In the complex plane, we represent points using complex numbers. Each point corresponds to a complex number of the form \( z = x + yi \).

For a circle with a center at a complex number \( z_1 = x_1 + y_1i \) with a radius \( r \), its equation is given by \( |z - z_1| = r \). This can be thought of as the distance between any point \( z \) on the circle and the center \( z_1 \), equating to the radius \( r \).

In our exercise, the center is at \( z_1 = 1 + 2i \) and the radius is 5, leading to the equation \( |z - z_1| = 5 \). Converting this into a general form equation involves squaring both sides and expanding the terms.

Distance Formula

The distance formula is a crucial concept for finding the distance between two points. In the context of complex numbers, it helps us compute the distance in the complex plane, similar to the Euclidean distance in Cartesian coordinates.

For any two points, represented by complex numbers \( z_1 = x_1 + y_1i \) and \( z_2 = x_2 + y_2i \), the distance \( d \) is given by the absolute value \( |z_2 - z_1| \), which simplifies to:

\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

In the problem, we have \( z = x + yi \) and \( z_1 = 1 + 2i \). Substituting into the formula, we get \( |z - z_1| = \sqrt{(x - 1)^2 + (y - 2)^2} \). This indicates that the set of all points \( z \) in this complex plane representation forms a circle with a radius equaling the fixed value, which in this case is 5 units.

Complex Plane

The complex plane is a fascinating visual way to represent complex numbers. Each complex number is uniquely represented as a point in this two-dimensional plane, where the horizontal axis is the real part and the vertical axis is the imaginary part of the number.

The origin, where both real and imaginary parts are zero, serves as a baseline. For any number \( z = x + yi \), the point sits at coordinates \( (x, y) \) in this plane. This provides a geometric interpretation of arithmetic operations on complex numbers.

In geometrical terms, the set \( |z - z_1| = r \) represents all points \( z \) that maintain a constant distance \( r \) from a specific point \( z_1 \), forming a circle. This visualization helps students connect algebraic expressions of complex numbers with their geometric figures.

Expanding Equations

Expanding equations is a fundamental technique, especially when working with algebraic forms in the context of circles and complex numbers. When you expand an equation of the form \( (x - x_1)^2 + (y - y_1)^2 = r^2 \), you break down each squared term using identities like \((a - b)^2 = a^2 - 2ab + b^2\).

In our problem, expanding the equation \( (x - 1)^2 + (y - 2)^2 = 25 \) results in:\( x^2 - 2x + 1 + y^2 - 4y + 4 = 25 \).

By simplifying the equation, we get \( x^2 - 2x + y^2 - 4y = 20 \). This equation matches one of the choices, and such expansions help us get into standard forms that might be needed for further analysis or comparison. The process ensures clarity in translating geometric problems in the complex plane to algebraic expressions.