Question

If \(|z|=1\) and \(w=[(z-1) /(z+1)](z \neq-1)\) then \(\operatorname{Re}(w)=\) (a) 0 (b) \(\left[1 /\left(|z+1|^{2}\right)\right]\) (c) \(\left[1 /\left(|z+1|^{3}\right)\right]\) (d) \(\left[\sqrt{2} /\left(|\mathrm{z}+1|^{2}\right)\right]\)

Short answer

(a) 0

Step-by-step solution

Express \(z\) as a complex number

Since \(z\) is a complex number with magnitude 1, we can write \(z\) as: \[z = \cos\theta + i\sin\theta\] where \(\theta\) is the argument of the complex number \(z\).

Substitute \(z\) into the expression of \(w\)

As \(z = \cos\theta + i\sin\theta\), we subsitute it into the expression of \(w\): \[w=\frac{(\cos\theta + i\sin\theta) - 1}{(\cos\theta + i\sin\theta) + 1}\]

Simplify the expression of \(w\)

Now, let's simplify the expression by performing the addition and subtraction inside the numerator and denominator separately: \[w = \frac{(\cos\theta - 1 + i\sin\theta)}{(\cos\theta + 1 + i\sin\theta)}\] Next, we multiply the numerator and denominator of the expression by the complex conjugate of the denominator \((\cos\theta + 1 - i\sin\theta)\): \[w = \frac{(\cos\theta - 1 + i\sin\theta)(\cos\theta + 1 - i\sin\theta)}{(\cos\theta + 1 + i\sin\theta)(\cos\theta + 1 - i\sin\theta)}\] This helps us to eliminate the imaginary part from the denominator.

Calculate the expression after multiplying by conjugates

Now, let's expand the expression and calculate both numerator and denominator: Numerator: \[((\cos\theta - 1)(\cos\theta + 1)) - (i\sin\theta)^2 + i\sin\theta((\cos\theta - 1)- (\cos\theta + 1))\] Denominator: \[(\cos\theta + 1)^2 + (\sin^2\theta - 2i\sin\theta(\cos\theta + 1))\]

Simplify the expression and extract the real part

Simplifying the expressions of the numerator and denominator: Numerator: \[(\cos^2\theta - 1) - \sin^2\theta + 2i\sin\theta\] Denominator: \[(\cos^2\theta + 2\cos\theta + 1) + (\sin^2\theta - 2i\sin\theta(\cos\theta + 1))\] Now, we notice that \(\cos^2\theta + \sin^2\theta = 1\) according to the trigonometric identity. Using this, we further simplify both numerator and denominator: Numerator: \[2i\sin\theta\] Denominator: \[2\cos\theta + 2 - 2i\sin\theta(\cos\theta + 1)\] We can now see that the real part of \(w\) is: \[\operatorname{Re}(w) = 0\] Therefore, the correct option is: (a) 0

Key concepts

Magnitude of a Complex Number

Understanding the magnitude of a complex number, often denoted by the absolute value symbol, is essential for students tackling problems with complex numbers, such as those found in competitive exams like JEE. A complex number is typically written in the form:
\( z = x + iy \),
where \(x\) is referred to as the real part, and \(y\) is the imaginary part of the complex number. The magnitude is the measure of the distance of the point representing the complex number on the complex plane from the origin (0,0). Mathematically, it's found by the formula:
\[ |z| = \sqrt{x^2 + y^2} \].
It's akin to the Pythagorean theorem where the magnitude is the hypotenuse of a right-angled triangle with sides \(x\) and \(y\) on the real and imaginary axes, respectively. This concept is fundamental in simplifying complex number expressions and finding the modulus in polar form.

Complex Number Argument

The argument of a complex number is as critical as its magnitude, especially when dealing with polar form or trigonometric representations of complex numbers. This angle—termed as the 'argument'—is the direction of the vector starting from the origin to the complex number when plotted on the complex plane.
To find the argument of a complex number \(z = x + iy\), one uses the formula:
\[ \theta = \arctan(\frac{y}{x}) \].
However, care must be taken with the quadrant in which the complex number resides to ensure the correct value of \(\theta\). The argument's principal value typically lies between \(-\pi\) and \(\pi\). In the context of competitive exams like JEE, understanding the argument helps to manipulate complex numbers in their exponential form \(z = re^{i\theta}\) where \(r\) is the magnitude, and \(\theta\) is the argument.

Real Part of Complex Number

The real part of a complex number is the component that resides on the real axis in the complex plane. Using the general form \(z = x + iy\), the real part is simply \(x\), and is symbolized as \(\operatorname{Re}(z)\). It can be any real number—including zero—which represents the horizontal distance from the origin.
When solving complex equations, particularly in examinations like JEE, separating the real part from the imaginary part often simplifies the process and yields a more straightforward solution pathway. As seen in the exercise, isolating the real part via manipulation, such as multiplying by the conjugate, is a fundamental technique. This enables us to solve for the real part independently or to equate real parts on both sides of an equation, as is often required in JEE problems.