Problem 105 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(|z-i|=1\) and \(\arg (z)=\theta, \theta \in\left(0, \frac{\pi}{2}\right)\), then the value of \(\cot \theta-\frac{2}{z}\) is equal to (A) 0 (B) \(i\) (C) \(-i\) (D) 1

Step-by-Step Solution

Answer

The value is (A) 0.

Step 1: Interpret the Given Condition

The condition \(|z-i|=1\) represents a circle centered at \(i\) with a radius of 1. In other words, any point \(z\) on this circle has a distance of 1 from the point \(i\).

Step 2: Express the Point on the Circle

Since the circle is centered at \(i\) (which is \(0+i\) in rectangular form), any point on this circle can be written as \(z=x+(y+i)i\) such that \((x, y+i)\) is on the circle. This means \((x, y)\) satisfies \((x-0)^2 + (y-1)^2 = 1^2\).

Step 3: Restrict \(\theta\) in the Given Range

We are given that \(\arg(z)=\theta\), and \(\theta \in (0, \frac{\pi}{2})\), which implies that \(z\) is in the first quadrant. Thus both the real part and imaginary part of \(z\) must be positive.

Step 4: Parameterize \(z\) on the Circle

Since \(z\) is on a circle of radius 1 centered at \(i\), the parameterization can be \(z = e^{i\phi} + i = \cos \phi + i(\sin \phi + 1)\) where \(\phi \in (0, \frac{\pi}{2})\) as adding \(i\) raises all imaginary parts by 1.

Step 5: Compute \(\cot \theta\)

From \(\theta = \arg(z)\), and the parameterization, \(\theta = \arg(\cos \phi + i(\sin \phi + 1))\), so \(\cot \theta = \frac{\text{Re}(z)}{\text{Im}(z)} = \frac{\cos \phi}{\sin \phi + 1}.\)

Step 6: Compute \(\frac{2}{z}\)

\(z = \cos \phi + i(\sin \phi + 1)\) implies \(\overline{z} = \cos \phi - i(\sin \phi + 1)\). Therefore, \(\frac{2}{z} = \frac{2(\cos \phi - i(\sin \phi + 1))}{(\cos \phi)^2 + (\sin \phi + 1)^2} = \frac{2(\cos \phi - i(\sin \phi + 1))}{1},\) as \((\cos \phi)^2 + (\sin \phi + 1)^2 = 2 \sin \phi + 2\).

Step 7: Simplify \(\cot \theta - \frac{2}{z}\) Expression

Combining results, the expression \(\cot \theta - \frac{2}{z}\) becomes \( \frac{\cos \phi}{\sin \phi + 1} - \frac{2\cos \phi}{2\sin \phi + 2} + \frac{2i(i(\sin \phi + 1))}{2\sin \phi + 2} = \frac{\cos \phi}{\sin \phi + 1} - \frac{\cos \phi + i i(\sin \phi + 1)}{\sin \phi + 1}.\) This ultimately simplifies to 0 after cancellation.

Key Concepts

Argand Diagram

The Argand Diagram is a visual representation of complex numbers. Picture this as a coordinate system where:

• The horizontal axis represents the real part of a complex number.
• The vertical axis represents the imaginary part.

On this diagram, any complex number, such as \(z = a + bi\), is represented as a point \((a, b)\) or a vector originating from the origin to the point \(a\) on the real axis, and \(b\) on the imaginary axis.
By using this coordinate-like system, it becomes incredibly easy to visualize and manipulate complex numbers geometrically.
The Argand Diagram is particularly useful in exercises like the one we are examining, where understanding the geometry of a problem is key. For example, in understanding a circle's condition \(|z-i|=1\), we can immediately picture a circle centered at \((0, 1)\) with a radius of 1 to find all possible values of \(z\). This visual insight helps in attaching real meaning to the conditions posed in complex plane problems.

Trigonometric Form of Complex Numbers

The trigonometric form of complex numbers is a method of expressing complex numbers in terms of their geometric relationships on the Argand Diagram.
This form is particularly helpful when dealing with rotations and scaling of complex numbers.
In the trigonometric form, a complex number \(z = a + bi\) is expressed as \(r (\cos \theta + i\sin \theta)\), where:

• \(r = \sqrt{a^2 + b^2}\) is the magnitude of the complex number (distance from the origin).
• \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\) is the argument (or angle), giving the direction of \(z\) from the positive real axis.

This form ties into our exercise through the parameterization of the circle, allowing us to articulate \(z\) as \(e^{i\phi} + i\), where \(\phi\) is adjusted to fit within the given angle's quadrant constraints.
It simplifies operations such as multiplication and division, which can then be understood and executed in terms of angle addition and subtraction.

Circle in Complex Plane

A circle in the complex plane is a set of complex numbers that satisfy a specific distance condition relative to a fixed point.
In our exercise, the condition \(|z - i| = 1\) defines a circle:

• The center of the circle is the point \(i\), which translates to \((0, 1)\) on the Argand Diagram.
• This circle includes all complex numbers \(z\) that maintain a fixed distance of 1 from its center.

The parameterization of points on the circle is crucial here. For instance, expressing \(z\) as \(\cos \phi + i(\sin \phi + 1)\) effectively describes any point lying on the circle as a function of the angle \(\phi\).
This geometric intuition is powerful since understanding circles in the complex plane helps to break down and solve expressions involving distances, magnitudes, and angles, such as those found in many introductory and advanced exercises in complex analysis.