Locus Of Complex Numbers Finding The Equation And Value Of A
In the fascinating realm of complex numbers, we often encounter geometric representations of these numbers. One such representation involves loci, which are sets of points that satisfy specific conditions. This article delves into a problem involving a complex number $u = 2 + ai$ that lies on a locus defined by the equation $|z - 3 + 2i| = |z - 6 + i|$. Our journey will involve deciphering the equation of the locus and subsequently determining the value of $a$.
Deciphering the Locus: A Geometric Interpretation
Let's first understand what the equation $|z - 3 + 2i| = |z - 6 + i|$ geometrically represents. In the complex plane, the expression $|z - w|$ gives the distance between the points representing the complex numbers $z$ and $w$. Therefore, the given equation states that the distance between the point representing $z$ and the point representing $3 - 2i$ is equal to the distance between the point representing $z$ and the point representing $6 - i$. This definition precisely describes the perpendicular bisector of the line segment joining the points $3 - 2i$ and $6 - i$ in the complex plane.
To further clarify, consider the points $A(3, -2)$ and $B(6, -1)$ in the complex plane. The locus of point $z$ satisfying the given condition is the set of all points equidistant from $A$ and $B$. This set forms a straight line that perpendicularly bisects the line segment $AB$. This line is the locus we seek to define.
To find the equation of this locus, we can use the distance formula. Let $z = x + yi$, where $x$ and $y$ are real numbers. The given equation becomes:
This can be rewritten as:
Using the definition of the modulus of a complex number, we get:
Squaring both sides to eliminate the square roots, we obtain:
Expanding the squares, we get:
Simplifying by canceling out the $x^2$ and $y^2$ terms and rearranging, we have:
Combining like terms, we get the equation of the locus:
Dividing both sides by 2, we obtain the simplified equation:
This is the equation of a straight line in the Cartesian plane, representing the perpendicular bisector of the line segment joining the points $A(3, -2)$ and $B(6, -1)$.
In summary, the equation $|z - 3 + 2i| = |z - 6 + i|$ represents a straight line in the complex plane. This line is the perpendicular bisector of the line segment connecting the points $3 - 2i$ and $6 - i$. The equation of this locus is 3x + y = 12, which we derived using the distance formula and algebraic manipulations. This detailed explanation provides a clear geometric and algebraic understanding of the locus represented by the given equation.
Determining the Value of 'a': A Substitution Approach
Now that we have determined the equation of the locus, we can proceed to find the value of $a$. We are given that the complex number $u = 2 + ai$ lies on this locus. This means that the coordinates of the point representing $u$ must satisfy the equation of the locus we found, which is $3x + y = 12$.
Since $u = 2 + ai$, we can identify the real part as $x = 2$ and the imaginary part as $y = a$. Substituting these values into the equation of the locus, we get:
This simplifies to:
Subtracting 6 from both sides, we find:
Therefore,
This straightforward substitution method allows us to determine the value of $a$ because we know the complex number $u$ lies on the locus, and we have already derived the equation of that locus. The value of $a$ is the y-coordinate of the point represented by the complex number $u$, and it must satisfy the equation of the line. This approach effectively combines geometric understanding with algebraic manipulation to solve the problem.
In conclusion, the value of a in the complex number $u = 2 + ai$ is 6. This was determined by substituting the real and imaginary parts of $u$ into the equation of the locus, 3x + y = 12. This demonstrates the practical application of the locus equation in finding specific values related to complex numbers lying on that locus. The process involves a direct substitution and a simple algebraic step, highlighting the elegance of the connection between complex numbers and their geometric representations.
Comprehensive Understanding of Loci and Complex Numbers
To truly grasp the problem we've solved, it's crucial to have a comprehensive understanding of loci in the context of complex numbers. A locus, in general, is the set of all points that satisfy a given condition. In the complex plane, this condition often involves the modulus of a complex number, which represents its distance from the origin or from another complex number.
The equation $|z - w| = r$, where $z$ is a variable complex number, $w$ is a fixed complex number, and $r$ is a real number, represents a circle in the complex plane. The center of this circle is the point representing the complex number $w$, and the radius is $r$. This is because the equation states that the distance between $z$ and $w$ is constant and equal to $r$.
In our problem, we encountered a slightly different type of equation: $|z - 3 + 2i| = |z - 6 + i|$. This equation, as we've seen, represents a straight line. The key difference here is that we have two moduli, and the equation states that the distances from $z$ to two fixed points are equal. This is the defining property of the perpendicular bisector of the line segment joining those two points.
The ability to interpret these equations geometrically is a fundamental skill in complex number theory. It allows us to visualize the problem and use geometric intuition to guide our algebraic manipulations. For example, in our case, recognizing that the equation represents a perpendicular bisector immediately gives us a clear picture of the locus and helps us understand why the solution involves finding the equation of a line.
The broader significance of understanding loci in complex numbers extends to various applications in mathematics, physics, and engineering. Complex numbers are used to represent alternating currents in electrical circuits, wave functions in quantum mechanics, and fluid flow in fluid dynamics. In each of these areas, loci can be used to describe the behavior of the system under different conditions. For instance, in electrical engineering, the locus of impedance in an AC circuit can provide valuable information about the circuit's resonance and stability.
Furthermore, the concept of loci is closely related to other geometric concepts such as transformations and mappings in the complex plane. Understanding how complex numbers transform under different mappings is essential for solving problems in complex analysis, a branch of mathematics that deals with functions of complex variables.
In summary, the concept of loci is a powerful tool in the study of complex numbers. It allows us to connect algebraic equations with geometric shapes, providing a visual and intuitive understanding of complex number relationships. This understanding is essential for solving a wide range of problems in mathematics and its applications, including the determination of values like 'a' in our problem and the analysis of complex systems in various fields. The ability to translate between algebraic and geometric representations is a hallmark of mathematical proficiency, and it is particularly valuable in the realm of complex numbers.
Conclusion: A Synthesis of Geometry and Algebra
In this exploration, we successfully navigated the complex plane to decipher the locus defined by $|z - 3 + 2i| = |z - 6 + i|$ and determine the value of $a$ in the complex number $u = 2 + ai$. Our journey highlighted the powerful interplay between geometry and algebra in the realm of complex numbers.
We began by interpreting the given equation geometrically, recognizing that it represents the perpendicular bisector of the line segment connecting the points $3 - 2i$ and $6 - i$ in the complex plane. This geometric insight provided a clear roadmap for our algebraic manipulations. We then translated the geometric condition into an algebraic equation by using the definition of the modulus of a complex number and the distance formula. This process led us to the equation of the locus: $3x + y = 12$.
Next, we leveraged the information that the complex number $u = 2 + ai$ lies on this locus. By substituting the real and imaginary parts of $u$ into the equation of the locus, we were able to solve for $a$, finding that $a = 6$. This step demonstrated the practical application of the locus equation in determining specific values related to complex numbers on the locus.
The key takeaway from this exercise is the importance of being able to move fluently between geometric and algebraic representations of complex numbers. The geometric interpretation provides intuition and guides the algebraic manipulations, while the algebraic representation allows for precise calculations and the determination of specific values.
Furthermore, this problem illustrates the broader significance of complex numbers in mathematics and its applications. Complex numbers provide a rich framework for representing and solving problems in various fields, including electrical engineering, quantum mechanics, and fluid dynamics. The concept of loci, as we've seen, is a fundamental tool in this framework, allowing us to describe the behavior of complex systems and analyze their properties.
In conclusion, the problem of finding the locus and the value of $a$ serves as a microcosm of the broader field of complex numbers. It demonstrates the power of combining geometric intuition with algebraic techniques to solve problems and provides a glimpse into the rich and interconnected world of mathematics. The successful resolution of this problem underscores the beauty and elegance of complex number theory and its importance in the mathematical landscape.
