Finding Locus And Value Of A For Complex Number U = 2 + Ai
Introduction to Complex Numbers and Loci
In the fascinating realm of mathematics, complex numbers extend the familiar number line into a two-dimensional plane, opening up a world of geometric interpretations and algebraic manipulations. A complex number is typically expressed in the form z = x + yi, where x and y are real numbers, and i is the imaginary unit, defined as the square root of -1. The beauty of complex numbers lies not only in their algebraic properties but also in their geometric representation on the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part. This allows us to visualize complex numbers as points on a plane and to explore their relationships using geometric concepts.
The locus of points is a fundamental concept in geometry, referring to the set of all points that satisfy a particular condition or equation. In the context of complex numbers, the locus often describes a geometric shape formed by complex numbers that fulfill a specific relationship. For example, the equation |z - c| = r, where z is a complex variable, c is a fixed complex number (the center), and r is a positive real number (the radius), represents a circle in the complex plane. This is because it describes all points z whose distance from the center c is equal to the radius r. Understanding loci is crucial for visualizing and interpreting complex number equations geometrically. It allows us to translate algebraic expressions into geometric shapes and vice versa, providing a powerful tool for solving problems and gaining deeper insights into the behavior of complex numbers.
The problem at hand involves finding the value of 'a' for the complex number u = 2 + ai, given that u lies on the locus of points defined by the equation |z - 3 + 2i| = |z - 6 + i|. This equation equates the distances from a variable complex number z to two fixed points in the complex plane, suggesting that the locus is a straight line – specifically, the perpendicular bisector of the line segment connecting the two fixed points. To solve this, we will first determine the equation of the locus by expressing the given condition in terms of the real and imaginary parts of z. Then, we will substitute u = 2 + ai into the equation and solve for 'a'. This process combines algebraic manipulation with geometric interpretation, highlighting the interconnectedness of these concepts in complex number theory. By working through this problem, we will reinforce our understanding of complex numbers, loci, and their applications.
Determining the Equation of the Locus
To unravel the mystery of the locus defined by |z - 3 + 2i| = |z - 6 + i|, we must first translate this equation into a more manageable algebraic form. The absolute value of a complex number represents its distance from the origin in the complex plane. Therefore, |z - 3 + 2i| represents the distance between the complex number z and the complex number 3 - 2i, while |z - 6 + i| represents the distance between z and 6 - i. The given equation states that these distances are equal. Geometrically, this means that the locus of z is the set of all points equidistant from the points 3 - 2i and 6 - i in the complex plane. This geometric interpretation immediately suggests that the locus is the perpendicular bisector of the line segment connecting these two points.
To find the equation of this locus, let's express the complex number z in its general form, z = x + yi, where x and y are real numbers. Substituting this into the given equation, we get |(x + yi) - 3 + 2i| = |(x + yi) - 6 + i|. Now, we can group the real and imaginary parts within the absolute value signs: |(x - 3) + (y + 2)i| = |(x - 6) + (y + 1)i|. The absolute value of a complex number a + bi is given by √(a² + b²). Applying this to both sides of the equation, we obtain √[(x - 3)² + (y + 2)²] = √[(x - 6)² + (y + 1)²].
To eliminate the square roots, we square both sides of the equation: (x - 3)² + (y + 2)² = (x - 6)² + (y + 1)². Expanding the squares, we have x² - 6x + 9 + y² + 4y + 4 = x² - 12x + 36 + y² + 2y + 1. Notice that the x² and y² terms appear on both sides and can be canceled out. This simplifies the equation significantly. After canceling and rearranging terms, we get -6x + 4y + 13 = -12x + 2y + 37. Moving all terms to one side, we obtain 6x + 2y - 24 = 0. Finally, we can divide the entire equation by 2 to simplify it further, resulting in the equation of the locus: 3x + y - 12 = 0. This is the equation of a straight line in the Cartesian plane, which confirms our earlier geometric intuition that the locus is a straight line. This equation represents all complex numbers z = x + yi that are equidistant from 3 - 2i and 6 - i. Now that we have the equation of the locus, we can proceed to find the value of 'a'.
Finding the Value of 'a'
Having successfully determined the equation of the locus as 3x + y - 12 = 0, our next step is to find the value of 'a' for the complex number u = 2 + ai. We are given that u lies on the locus, which means that the coordinates of u must satisfy the equation of the locus. To apply this, we recognize that the real part of u is 2, and the imaginary part is a. In the context of the locus equation, the real part corresponds to x, and the imaginary part corresponds to y. Therefore, we can substitute x = 2 and y = a into the equation of the locus.
Substituting these values into 3x + y - 12 = 0, we get 3(2) + a - 12 = 0. This simplifies to 6 + a - 12 = 0. Now, we can solve for a. Combining the constant terms, we have a - 6 = 0. Adding 6 to both sides of the equation, we find a = 6. This is the value of 'a' that makes the complex number u = 2 + ai lie on the locus defined by |z - 3 + 2i| = |z - 6 + i|. Geometrically, this means that the point representing the complex number 2 + 6i in the complex plane is equidistant from the points representing 3 - 2i and 6 - i.
Thus, we have successfully found the value of 'a' by utilizing the equation of the locus. This process demonstrates the interplay between the algebraic representation of complex numbers and their geometric interpretation. By translating the given condition into an algebraic equation and then substituting the complex number u, we were able to solve for the unknown variable 'a'. This highlights the power of combining algebraic and geometric techniques in solving complex number problems. The value a = 6 is the unique solution that satisfies the given conditions, placing the complex number u on the specified locus.
Conclusion: Unveiling the Interplay of Algebra and Geometry
In conclusion, we have successfully navigated the problem of finding the value of 'a' for the complex number u = 2 + ai residing on the locus defined by |z - 3 + 2i| = |z - 6 + i|. This journey has underscored the profound connection between algebraic manipulation and geometric interpretation in the realm of complex numbers. We began by recognizing that the given equation represents the set of all points z in the complex plane that are equidistant from the points 3 - 2i and 6 - i, thus geometrically defining a straight line – the perpendicular bisector of the segment connecting these two points.
Our approach involved a methodical translation of the geometric condition into an algebraic equation. By expressing z as x + yi and applying the definition of the absolute value of a complex number, we transformed the initial equation into an algebraic form. Squaring both sides to eliminate the square roots and simplifying the resulting expression, we derived the equation of the locus as 3x + y - 12 = 0. This equation represents a straight line in the Cartesian plane, confirming our geometric intuition. The process of deriving this equation showcased the power of algebraic manipulation in representing geometric relationships.
Subsequently, we leveraged the fact that the complex number u = 2 + ai lies on the locus. This implied that the coordinates of u must satisfy the equation of the locus. Substituting x = 2 and y = a into the equation 3x + y - 12 = 0, we obtained a simple linear equation in terms of 'a'. Solving this equation, we determined that a = 6. This value of 'a' uniquely positions the complex number u on the specified locus, making it equidistant from the points 3 - 2i and 6 - i. The solution a = 6 is a testament to the precision and elegance of complex number theory.
This problem serves as a compelling illustration of how algebraic and geometric perspectives complement each other in mathematics. The ability to translate between these perspectives is a powerful tool for solving problems and deepening our understanding of mathematical concepts. The journey from the initial equation to the final solution highlights the interconnectedness of different branches of mathematics and the importance of a holistic approach to problem-solving. By mastering the interplay between algebra and geometry, we can unlock new insights and tackle complex challenges with confidence and creativity.
