Problem 48 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(P, P^{\prime}\) represent the complex number \(z_{1}\) and its additive inverse respectively then the complex equation of the circle with \(P P^{\prime}\) as a diameter is (A) \(\frac{z}{z_{1}}=\overline{\left(\frac{z_{1}}{z}\right)}\) (B) \(z \bar{z}+z_{1} \bar{z}_{1}=0\) (C) \(z \bar{z}_{1}+\bar{z} z_{1}\) (D) None of these

Step-by-Step Solution

Answer

The answer is (D) None of these.

Step 1: Understanding the problem

We need to derive the equation of the circle for which the complex number \( z_1 \) and its additive inverse \( z_1' \) are diametric endpoints. This involves applying the formula for the equation of a circle in the complex plane given the endpoints of its diameter.

Step 2: Use the diameter endpoints

The general complex form of a circle with diameter endpoints \( z_1 \) and \( z_1' \) is given by the equation \( |z - z_1| \cdot |z - z_1'| = r^2 \), where \( r \) is the radius. From the context of the problem, \( P' = -z_1 \) since it is the additive inverse.

Step 3: Apply mid-point formula

The center of the circle is the midpoint of the diameter. Thus, \[ C = \frac{z_1 + (-z_1)}{2} = 0. \] So, the center is at \( z = 0 \), and the radius is \( |z_1 - (-z_1)|/2 = |2z_1|/2 = |z_1|. \)

Step 4: Equation of the circle

Using the circle with center at 0 and radius \( |z_1| \), the equation is written as:\[ |z| = |z_1|. \] This can be rewritten in terms of squares as:\[ z \bar{z}= |z_1|^2. \]

Step 5: Final comparison

None of the given options correspond directly to \( z \bar{z} = |z_1|^2 \) or \( z \bar{z} - z_1 \bar{z}_1 = 0 \). Therefore, the correct answer is (D) None of these.

Key Concepts

The Circle Equation in the Complex Plane

In the complex plane, circles can be represented using complex numbers. The unique property of using complex numbers for representing circles lies in how elegantly they capture geometric properties. The general equation of a circle with a given center and radius in the complex plane is

• If the center is a point represented by a complex number \(c\) and the radius is \(r\), then the equation of a circle is given by: \(|z - c| = r\).

This equation simply states that all points \(z\) on the circle are a distance \(r\) from the center \(c\).

In our exercise, the center of the circle lies at the origin \(0\) due to the properties of the points being additive inverses. Consequently, the equation simplifies to \(|z| = r\), and by squaring both sides, it becomes \(z \bar{z} = r^2\), capturing the full geometric significance of the circle.

Understanding the Additive Inverse in Complex Numbers

The concept of the additive inverse is fundamental in understanding symmetries in complex numbers, especially in geometric constructions.

• For any complex number \(z_1 = a + bi\), its additive inverse is \(-z_1 = -a - bi\).
• Graphically, the point and its additive inverse are reflections of each other through the origin.
• This property is essential for defining the diameter of a circle whose endpoints are \(z_1\) and its additive inverse \(P' = -z_1\).

The midpoint of these two complex numbers becomes zero, which simplifies computations by ensuring the center of the circle falls right on the origin.

Knowing this property allows us to find that \(|z_1 + (-z_1)|/2 = 0\), confirming that we are indeed using the center at the origin when forming the circle equation.

Using Diameter Endpoints to Define a Circle

To form the equation of a circle using the diameter endpoints in the complex plane, pivotal geometric insights arise.

The important steps are:

• Identify the endpoints of the diameter, for example, \(z_1\) and \(-z_1\).
• Calculate their midpoint to find the circle's center. For the points in our exercise, this midpoint is \(0\), making it straightforward.
• Calculate the radius by considering the distance between the endpoints. The radius here would be half the length of the diameter, or \(|z_1|\).

With these elements, the full circle equation is derived as \(|z| = |z_1|\). By expressing this as \(z \bar{z} = |z_1|^2\), we neatly summarize the circle's properties using complex numbers.

Thus, these computations underline the geometric intuition that the circle has its center at the midpoint of the diameter and uses half its length as the radius.