Problem 68 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Two circles in complex plane are \(C_{1}:|z-i|=2\) \(C_{2}:|z-1-2 i|=4\). Then (A) \(C_{1}\) and \(C_{2}\) touch each other. (B) \(C_{1}\) and \(C_{2}\) intersect at two distinct points. (C) \(C_{1}\) lies within \(C_{2}\). (D) \(C_{2}\) lies within \(C_{1}\) -

Step-by-Step Solution

Answer

(C) \(C_1\) lies within \(C_2\).

Step 1: Identify Center and Radii

The equation for a circle in the complex plane is \(|z - a| = r\), where \(a\) is the center and \(r\) is the radius. For \(C_1\), the center is \(i\) (or \(0 + i\)) and the radius is \(2\). For \(C_2\), the center is \(1 + 2i\) and the radius is \(4\).

Step 2: Calculate Distance Between Centers

To find the distance between the centers \(i\) and \(1 + 2i\), use the distance formula. The difference between the centers is \((1 + 2i) - i = 1 + i\). The magnitude is \(|1 + i| = \sqrt{1^2 + 1^2} = \sqrt{2}\).

Step 3: Compare Distance with Sum and Difference of Radii

The sum of the radii is \(2 + 4 = 6\), and the difference is \(4 - 2 = 2\). Comparing the distance \(\sqrt{2} \approx 1.41\) to these values, since \(\sqrt{2}\) is less than \(2\), \(C_1\) lies entirely inside \(C_2\) without touching the boundary of \(C_2\).

Step 4: Determine Relationship

Since the distance between the centers \(\sqrt{2}\) is less than the difference in radii, \(C_1\) is completely within \(C_2\). Therefore, option (C) "\(C_1\) lies within \(C_2\)" is true.

Key Concepts

Complex Plane

The complex plane is a two-dimensional plane used to graph complex numbers. It is similar to a Cartesian coordinate system. Here, the x-axis represents the real part of a complex number, and the y-axis represents the imaginary part. A complex number, such as \( z = a + bi \), can be represented as a point \(a, b\) on the complex plane.
Understanding the layout of the complex plane helps when analyzing transformations, geometric locations, and distances between complex numbers. By visualizing numbers in this format, tasks like adding, subtracting, and calculating distances become more intuitive.

Circle Equations

In the context of the complex plane, a circle can be represented by the equation \( |z - a| = r \), where \(a\) is the center of the circle, and \(r\) is the radius. The vertical bar represents the modulus, or absolute value, of the complex number, which translates to a measure of distance from the center.
This particular equation helps in identifying the geometric properties of a circle in the complex plane. For example, given the circles \(C_1:|z-i|=2\), and \(C_2:|z-1-2i|=4\), \(C_1\) has its center at \(i\) with a radius of \(2\), while \(C_2\) is centered at \(1 + 2i\) with a radius of \(4\). Knowing this makes it easier to plot these circles and understand their placement relative to each other.

Distance Formula

Calculating the distance between two points in the complex plane uses the distance formula, which is derived from the modulus of a complex number. If you have two complex numbers, say \(z_1 = a + bi\) and \(z_2 = c + di\), the distance between them is given by \( |z_2 - z_1| = \sqrt{(c-a)^2 + (d-b)^2} \).
In our example, the centers \(i\) and \(1 + 2i\) give us a distance of \(\sqrt{2}\), which reflects the ease with which geometric properties such as proximity or intersection can be analyzed.

Geometric Properties

The geometric properties of circles in the complex plane are crucial in understanding their relationships. These properties include their centers, radii, and how they interact with each other. Two circles can intersect, be tangent, or one may lie within the other.
In this exercise, we calculated the distance between the centers of \(C_1\) and \(C_2\) as \(\sqrt{2}\). Since this distance is less than the difference between the radii \(4 - 2 = 2\), \(C_1\) is entirely within \(C_2\). Knowing how to compare these distances with the sums and differences of radii allows for a clear understanding of circle containment and other geometric properties in the complex plane.