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

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 the centers and radii of the circles

For circle \(C_1\), the general form is \(|z - z_1| = r_1\), where the center \(z_1 = i\) (or \(0 + i\)) and the radius \(r_1 = 2\). For circle \(C_2\), \(|z - z_2| = r_2\), with center \(z_2 = 1 + 2i\) and radius \(r_2 = 4\).

Step 2: Calculate the distance between the centers

The distance \(d\) between the centers \((0, 1)\) and \((1, 2)\) is calculated using the distance formula: \[ d = \sqrt{(1-0)^2 + (2-1)^2} = \sqrt{1 + 1} = \sqrt{2} \approx 1.414. \]

Step 3: Analyze the relation between distance and radii

Compare the distance \(d = \sqrt{2}\) with the sum of the radii \(r_1 + r_2 = 2 + 4 = 6\) and the absolute difference of the radii \(|r_1 - r_2| = |2 - 4| = 2\).

Step 4: Determine the configuration of the circles

Since \(\sqrt{2} < 2\), the distance between the centers is less than the difference of the radii. Therefore, circle \(C_1\) lies entirely within circle \(C_2\) without touching it.

Key Concepts

Circle Equations

In the realm of complex plane geometry, understanding circle equations effectively is key. A circle can be expressed in the complex plane using the standard form

• \(|z - z_0| = r\)

where \(z\) denotes any point on the circle, \(z_0\) represents the center of the circle, and \(r\) stands for the radius.
This form is incredibly convenient for calculations. For example, circle \(C_1\) in the exercise is defined by \(|z - i| = 2\), where \(z_0 = i\) (or \(0 + i\)), and the radius is 2. This equation tells us exactly how far any point on the circle is from the center.
Likewise, circle \(C_2\) is described by \(|z - (1 + 2i)| = 4\), giving it a center of \(1 + 2i\) and a radius of 4. This straightforward notation allows mathematicians to quickly visualize and solve geometric problems.

Distance Between Centers

A critical step in analyzing circle configurations is determining the distance between the centers of the circles. This distance can be thought of as a direct line connecting the centers, calculated with the formula:

• \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

Here, \(x_1, y_1\) and \(x_2, y_2\) are the real and imaginary components of each center, respectively.
For circle \(C_1\) centered at \(i\) (which translates to \((0, 1)\) in the complex plane), and circle \(C_2\) centered at \(1 + 2i\) (thus \((1, 2)\)), the distance between their centers becomes \(\sqrt{2}\). This calculation is vital as it guides our understanding of how the circles interact with each other.

Circle Configuration

The configuration of two circles depends on both the distance between their centers and their radii. Whether two circles touch, intersect, or one lies inside the other is determined through comparison of these measurements.
The relationship among these properties can be summed up in different possible scenarios like:

• If \(d = r_1 + r_2\), the circles touch each other externally.
• If \(d = |r_1 - r_2|\), they touch internally.
• If \(d < |r_1 - r_2|\), one circle lies completely inside the other.
• If \(d > r_1 + r_2\), the circles are separate.

In our exercise, since \(\sqrt{2} < 2\) (and much less than 6), it indicates that circle \(C_1\) lies entirely within circle \(C_2\) without any points of contact. Hence, they don't intersect or touch each other, providing clarity on their spatial relationship.

Complex Numbers

Complex numbers are the backbone of many areas in mathematics, including geometry in the complex plane. A complex number takes the form \(z = x + yi\), where \(x\) is the real part and \(y\) is the imaginary part.
Their beauty lies in how they simplify two-dimensional geometric problems into algebraic equations. Each complex number corresponds to a point on the plane; here, the centers of our circles, \(i\) and \(1+2i\), are examples of such points.
These numbers make it easier to calculate distances and interpret relationships between different sets of points (or circles here) with an elegant fusion of geometry and algebra. Understanding complex numbers allows one to elegantly and precisely solve geometric problems that involve spatial reasoning in two dimensions.