Problem 151
Question
The locus represented by the complex equation \(|z-2-i|=|z| \sin \left(\frac{\pi}{4}-\arg z\right)\) is the part of (A) a pair of straight lines (B) a circle (C) a parabola (D) a rectangular hyperbola
Step-by-Step Solution
Verified Answer
The locus is a rectangular hyperbola.
1Step 1: Understanding the Locus Equation
The given complex equation is \(|z-2-i| = |z| \sin\left(\frac{\pi}{4} - \arg z\right)\). Let \(z = x + yi\), so that we can rewrite \(|z| = \sqrt{x^2 + y^2}\) and \(\arg z = \tan^{-1}(\frac{y}{x})\). The other side, \(|z-2-i|\), simplifies to \(\sqrt{(x-2)^2 + (y-1)^2}\). The goal is to understand the geometric figure this equation represents.
2Step 2: Analyzing Modulus Relation
Examine the expression \(|z-2-i| = |z| \sin\left(\frac{\pi}{4} - \arg z\right)\). Convert \(\sin\left(\frac{\pi}{4} - \arg z\right)\) to \(\sin(\frac{\pi}{4})\cos(\arg z) - \cos(\frac{\pi}{4})\sin(\arg z)\). Simplifying, this is \(\frac{1}{\sqrt{2}}(\frac{x}{|z|} - \frac{y}{|z|})\). Substitute back into the given equation for substitution.
3Step 3: Simplifying the Equation
Substitute \(|z| = \sqrt{x^2 + y^2}\) into the equation giving \(|z-2-i| = \frac{1}{\sqrt{2}}(x-y)\). The equation is then \(\sqrt{(x-2)^2 + (y-1)^2} = \frac{x-y}{\sqrt{2}\sqrt{x^2 + y^2}}\). This simplifies to a quadratic form upon squaring, aiming to isolate xy terms if any.
4Step 4: Deriving the Locus Type
Solve \((x-2)^2 + (y-1)^2=(x-y)^2 = (x-y)^2\) by breaking down into expansions and simplifying to show a quadratic equation. Rearranging parameters via squaring both sides and cross-multiplying, observe the equation formation.
5Step 5: Identifying the Conic Section
Notice that from step 4, if there is a constant crossed variable term or zero linear term, it highly resembles forms of hyperbolas. The equation has characteristics of a rectangular hyperbola, noted when equated to specific conic forms without need of completing the square, confirming geometric figure.
Key Concepts
LocusRectangular HyperbolaComplex Numbers
Locus
The term "locus" refers to a set of points that satisfy a particular condition or equation. Imagine drawing shapes by plotting points that meet certain criteria; this is what a locus represents, essentially a blueprint for a shape.
In complex geometry, working with loci can involve interpreting relationships of points on a complex plane. When using complex numbers, loci often involve equations with moduli, arguments, and real and imaginary components.
For example, in the equation \(|z-a| = |z| \sin\left(\frac{\pi}{4} - \arg z\right)\), the locus is defined by the condition relating the distance from a fixed point \(a\) and another property of \(z\).
Understanding loci in complex numbers helps clarify the movement and positioning of shapes in complex geometry, offering a bridge between algebraic expressions and geometric intuition.
In complex geometry, working with loci can involve interpreting relationships of points on a complex plane. When using complex numbers, loci often involve equations with moduli, arguments, and real and imaginary components.
For example, in the equation \(|z-a| = |z| \sin\left(\frac{\pi}{4} - \arg z\right)\), the locus is defined by the condition relating the distance from a fixed point \(a\) and another property of \(z\).
Understanding loci in complex numbers helps clarify the movement and positioning of shapes in complex geometry, offering a bridge between algebraic expressions and geometric intuition.
Rectangular Hyperbola
A rectangular hyperbola is a special type of conic section. Unlike standard hyperbolas, which appear elongated and have asymptotes forming an angle not equal to \(90^\circ\), a rectangular hyperbola's asymptotes are perpendicular to each other.
A rectangular hyperbola has the general equation \(xy = c^2\), where \(c\) is a constant. This equation means that moving along the hyperbola, the product of the coordinates remains constant.
Rectangular hyperbolas are fascinating in complex geometry, as they can represent relationships where altering one variable requires balancing the others. They're particularly striking because they visually represent key algebraic identities and maintain symmetry despite the challenging nature of hyperbolic curves.
In analyzing complex equations, spotting the characteristics of a rectangular hyperbola can reveal deeper insights into the nature of solutions, offering a window into geometric harmony hidden in algebraic expressions.
A rectangular hyperbola has the general equation \(xy = c^2\), where \(c\) is a constant. This equation means that moving along the hyperbola, the product of the coordinates remains constant.
Rectangular hyperbolas are fascinating in complex geometry, as they can represent relationships where altering one variable requires balancing the others. They're particularly striking because they visually represent key algebraic identities and maintain symmetry despite the challenging nature of hyperbolic curves.
In analyzing complex equations, spotting the characteristics of a rectangular hyperbola can reveal deeper insights into the nature of solutions, offering a window into geometric harmony hidden in algebraic expressions.
Complex Numbers
Complex numbers are an extension of the real number system and incorporate an extra dimension for solving equations that don't have real solutions. Written in the form \(z = x + yi\), they consist of a real part \(x\) and an imaginary part \(y\).
The imaginary unit \(i\) satisfies \(i^2 = -1\), allowing complex numbers to address equations like \(x^2 + 1 = 0\), which lack real solutions.
Using complex numbers we can explore different geometric transformations, plot curves, and panels in complex geometry. Essential operations such as taking moduli, interpreting arguments, and transforming them in the complex plane lead to insights into otherwise inaccessible geometric forms.
Complex numbers are very useful in expressing problems involving oscillations and waves, electrical engineering, and quantum physics, linking algebraic formulas and physical phenomena in a unified context.
The imaginary unit \(i\) satisfies \(i^2 = -1\), allowing complex numbers to address equations like \(x^2 + 1 = 0\), which lack real solutions.
Using complex numbers we can explore different geometric transformations, plot curves, and panels in complex geometry. Essential operations such as taking moduli, interpreting arguments, and transforming them in the complex plane lead to insights into otherwise inaccessible geometric forms.
Complex numbers are very useful in expressing problems involving oscillations and waves, electrical engineering, and quantum physics, linking algebraic formulas and physical phenomena in a unified context.
Other exercises in this chapter
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