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Chapter 3: Problem 26

Determine the inverse point of \(1+i\) with respect to the circle \(|z+1-2 i|=2\)

Short Answer

Expert verified
The inverse point of \(1+i\) with respect to the circle \(|z+1-2i| = 2\) is \(-3 - 2i\).

Step by step solution

01

Identify the Center and Radius of the Given Circle

First identify the center and radius of the circle \(|z+1-2 i|=2\). We rewrite the equation by isolating z: \(z = -1 + 2i + z_1\), where \(z_1\) is any complex number on the circle. This gives us the center of the circle (p) as -1+ 2i, and the radius (r) as 2.
02

Apply the Inversion Formula with the Given Data

Substitute the identified complex number `z = 1+i`, the center of the circle `p = -1+2i`, and radius `r = 2`, into the inversion formula. This gives: \[z' = (-1 + 2i) + \frac{(2)^2}{1+i - (-1 + 2i)}\]
03

Simplify the Resulting Expression

Simplify the equation to get the result 鈥� \[z' = - 1 + 2i + \frac{4}{2+ i}\]. Further simplification gives the inverse point to be \( z' = -3 - 2i\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inverse Point
Understanding the concept of an inverse point is crucial in complex analysis, especially when dealing with transformations. When you have a specific point in the complex plane and a circle, the inverse point of that location relative to the circle is another point such that both lie on a line through the circle's center. This idea is often tied to circle inversion, where the point inverts in relation to the circle, flipping inside and outside its boundary.
Complex Numbers
Complex numbers are foundational in mathematics, representing numbers that can be expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit, satisfying \(i^2 = -1\). These numbers are plotted on a complex plane, with the real part \(a\) on the x-axis and the imaginary part \(b\) on the y-axis.
Understanding complex numbers allows you to perform operations such as addition, subtraction, multiplication, and division, similar to real numbers, but with additional rules involving \(i\). This versatility makes complex numbers powerful in solving equations that have no real solutions.
Circle in Complex Plane
In the complex plane, a circle is defined quite similarly to the Cartesian plane, but with complex coordinates. A circle's equation is typically given by \(|z-p| = r\), where \(p\) is the circle's center, and \(r\) is the radius.
  • \(p\) is a complex number representing the center, including both real and imaginary parts.
  • \(r\) is the radius, defining the distance from the center to any point on the circle.
Manipulating this equation allows us to explore transformations and relationships among complex points in intricate ways, such as finding inverse points.
Inversion Formula
The inversion formula is a mathematical expression used to derive the inverse point of a given point with respect to a circle. It is typically expressed as:
\[ z' = p + \frac{r^2}{z - p} \]
  • \(z\) is the complex number representing the point you wish to invert.
  • \(p\) is the center of the circle.
  • \(r\) is the radius of the circle.
The formula rearranges the relationship of the point with the circle, effectively swapping its position and distance relative to the circle's center. Mastering this formula is essential for anyone dealing with complex transformations or seeking to understand the deeper geometry in complex analysis.

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Most popular questions from this chapter

Find a bilinear transformation \(f\) which maps the circle \(|z+i|=1\) onto the real line \(\mathbb{R}\).

Prove that the cross ratio of four distinct points is real if and only if the four points lie on a circle or on a straight line.

Show that a bilinear transformation has either 1,2 or infinitely many fixed points. Establish conditions for each occurrence.

Prove that the bilinear transformation $$ w=T(z)=\frac{\left(z_{1}+z_{2}\right) z-2 z_{1} z_{2}}{2 z-\left(z_{1}+z_{2}\right)} \quad\left(z_{1} \neq z_{2}\right) $$ has the fixed points \(z_{1}\) and \(z_{2}\), and show that \(T(T(z))=z\).

For the mapping \(w=(1+i) z+2\), find the image of (a) the line \(y=2 x\) (b) the line \(y=3 x+2\) (c) the circle \(|z|=3\) (d) the circle \(|z-1|=2\).
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