Question

Let \(a, b \in \mathbb{R}\) and \(a^{2}+b^{2} \neq 0 .\) Suppose \(S=\left\\{z \in \mathbb{C}: z=\frac{1}{a+i b t}, t \in \mathbb{R}, t \neq 0\right\\}\), where \(i=\sqrt{-1}\). If \(z=x+i y\) and \(z \in S\), then \((x, y)\) lies on (A) the circle with radius \(\frac{1}{2 a}\) and centre \(\left(\frac{1}{2 a}, 0\right)\) for \(a>0, b \neq 0\) (B) the circle with radius \(-\frac{1}{2 a}\) and centre \(\left(-\frac{1}{2 a}, 0\right)\) for \(a<0, b \neq 0\) (C) the \(x\) -axis for \(a \neq 0, b=0\) (D) the \(y\) -axis for \(a=0, b \neq 0\)

Short answer

Choice (A): the locus of (x, y) is the circle with radius \(\frac{1}{2a}\) and centre \(\left(\frac{1}{2a}, 0\right)\) for \(a>0, b eq 0\).

Step-by-step solution

Express z

Write the complex number z from set S as z using the given condition \(z = \frac{1}{a + ibt}\)

Apply complex conjugation

Multiply the numerator and the denominator of z by the complex conjugate of the denominator to remove the complex term from the denominator: \(z = \frac{1}{a + ibt} \cdot \frac{a -ibt}{a -ibt}\).

Simplify the expression

Carry out the multiplication to get \(z = \frac{a - ibt}{a^2 + b^2t^2}\).

Separate the real and imaginary parts

Express z in terms of its real and imaginary parts: \(z = \frac{a}{a^2 + b^2t^2} + i\left(\frac{-bt}{a^2 + b^2t^2}\right)\).

Compare with standard coordinates

Recognize that the real part of z represents x and its imaginary part (multiplied by i) represents y: \(x = \frac{a}{a^2 + b^2t^2}\) and \(y = \frac{-bt}{a^2 + b^2t^2}\).

Analyze the cases for the real number a and b

Analyze each provided option by comparing with the expressions for x and y and considering the conditions for a and b, and seeking for the description of the locus of (x, y).

Conclusion

Use the derived expressions to determine that the correct choice is (A) since the expressions for x and y describe a circle with radius \(\frac{1}{2a}\) and center \(\left(\frac{1}{2a}, 0\right)\) provided that \(a>0\) and \(b eq 0\), as squaring x and y and adding them yields \(\frac{a^2}{(a^2+b^2t^2)^2} + \frac{b^2t^2}{(a^2+b^2t^2)^2} = \frac{1}{(2a)^2}\), which is the equation of a circle.

Key concepts

Complex conjugation

Understanding complex conjugation is essential in simplifying expressions involving complex numbers and locating points related to these numbers on the complex plane. Complex conjugation is the process of changing the sign of the imaginary part of a complex number. If we have a complex number such as z = a + ib, where a and b are real numbers, and i is the imaginary unit, its complex conjugate is denoted as \(\bar{z} = a - ib\).

When dealing with fractions that include complex numbers in the denominator, as shown in the step-by-step solution, we utilize complex conjugation to 'rationalize' the denominator. Multiplying the numerator and denominator by the complex conjugate of the denominator helps remove the imaginary part, resulting in a fraction with a real denominator. This allows us to separate the complex number z into its real (x) and imaginary (y) components, which are crucial to plotting points or determining loci in the complex plane.

Locus of complex number

The term 'locus' refers to a set of points satisfying a particular condition or a set of conditions. In terms of complex numbers, the locus represents the path or figure created by a complex number z when it is plotted in the complex plane, with each point corresponding to certain values of the real part x and the imaginary part y.

In the given exercise, we are concerned with the locus of the complex number z belonging to the set S, as the parameter t varies. By simplifying z and plotting the resulting x and y values, we can determine the shape and position of this locus. For instance, depending on the values of a and b, the locus may form a circle, or line up along an axis as indicated in the original problem statement. This illustrates how the concept of locus helps in visualizing complex numbers in the geometric context of the complex plane.

Circle equation in complex plane

The equation of a circle in the complex plane is represented in the standard Cartesian form \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle and r is its radius. To find the equation of a circle from a complex function, we can relate the real and imaginary parts of the complex function to the coordinates x and y respectively.

Taking the example given in the original exercise, once we've separated the real and imaginary parts of the complex number z and equated them to x and y, we can square these values and sum them up to derive the equation of a circle. If the circle's equation aligns with the aforementioned standard form, we can then identify the center and radius of the circle. This is precisely what was done in the solution to correctly select option (A), portraying how mathematical manipulation of complex numbers can lead us to geometric interpretations on the complex plane.