Let \(a, b, x\) and \(y\) be real numbers such that \(a-b=1\) and \(y \neq 0\). If the complex number \(z=x+i y\) satisfies \(\operatorname{Im}\left(\frac{a z+b}{z+1}\right)=\mathrm{y}\), then which of the following is(are) possible value(s) of \(x ?\) [A] \(-1+\sqrt{1-y^{2}}\) [B] \(-1-\sqrt{1-y^{2}}\) [C] \(1+\sqrt{1+y^{2}}\) [D] \(1-\sqrt{1+y^{2}}\)

The imaginary part of a complex number is critical to understanding complex number algebra. When dealing with complex numbers, such as in the expression \( z = x + iy \), the 'imaginary part' refers to the coefficient of the imaginary unit \(i\) — in this case, \(y\). Imaginary units are defined by the property that \(i^2 = -1\).

The given exercise involves isolating the imaginary part after manipulating the complex expression \( \frac{a z + b}{z + 1} \). The operation \( \operatorname{Im}(expression) \) signifies we are only interested in the coefficient of the imaginary unit \(i\) that appears after performing the complex algebra.

When working with complex numbers like \( z = x + iy \), it's essential to understand how to perform algebraic operations. In our problem, we are given an expression that mixes complex and real numbers, and one of our tasks is to simplify it using algebraic manipulation.

In this process, we first substitute known values, and then we strategically use the property that \( a - b = 1 \). Simplifying the expression and separating the real and imaginary parts is key. It's also important to multiply and divide by the complex conjugate, which, for a complex number \( z \), is \( \bar{z} = x - iy \) to eliminate the imaginary unit from the denominator. This step stabilizes the equation and sets up the next phase of the solution—extracting the imaginary part for comparison with the given \( y \). Through this process, students develop a deeper understanding of the operations involving complex numbers.

The Joint Entrance Examination (JEE) Advanced poses challenging problems that evaluate a student's grasp of various mathematical concepts—including those of complex numbers. To excel in JEE problem-solving, students should focus on a systematic approach: start by understanding the problem, identify what is known and what is to be found, and carefully carry out step-by-step algebraic manipulations.

For instance, in this problem, we are tasked with finding possible values of \( x \) given a condition related to the imaginary part of a complex expression. The solution involves patience and precision—substituting values, simplification, and thoughtful consideration in equating the imaginary parts. It encapsulates the JEE exam's objective of testing a student's ability to apply mathematical theory to complex and multifaceted problems.