Problem 205 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(|z+4| \leq 3\), then the maximum value of \(|z+1|\) is [2007| (A) 4 (B) 10 (C) 6 (D) 0

Step-by-Step Solution

Answer

The maximum value of \(|z+1|\) is 6, which corresponds to option (C).

Step 1: Understand the inequality

Consider the inequality \(|z + 4| \leq 3\). This represents all complex numbers \(z\) such that their distance from \(-4\) in the complex plane is less than or equal to 3. This forms a circle centered at \(-4\) with radius 3.

Step 2: Express z in terms of real numbers

Let \(z = x + yi\), where \(x\) and \(y\) are real numbers. So \(|z + 4| = |x + 4 + yi| = \sqrt{(x + 4)^2 + y^2}\). We need \(\sqrt{(x + 4)^2 + y^2} \leq 3\).

Step 3: Convert the inequality to an equation

Square both sides to get \((x + 4)^2 + y^2 \leq 9\). This is the equation of a circle centered at \(-4, 0\) with radius 3.

Step 4: Find expression for |z+1|

The expression \(|z+1|\) is equivalent to \(\sqrt{(x+1)^2+y^2}\). Our task is to maximize this value given that \( (x + 4)^2 + y^2 \leq 9 \).

Step 5: Analyze the problem geometrically

We are looking for the point within or on the boundary of the circle \((x + 4)^2 + y^2 = 9\), such that the distance from the point to \(-1, 0\) is maximized.

Step 6: Determine the maximum distance

The furthest point in the circle from \(-1, 0\) will be on the line connecting \(-4, 0\) and \(-1, 0\). This point is \(-7, 0\), which is exactly opposite to \(-1, 0\) along the diameter. The distance between \(-7, 0\) and \(-1, 0\) is \(|-1+7| = 6\).

Step 7: Finalize the maximum value

The maximum value of \(|z+1|\) is achieved when \(z = -7 + 0i\), which gives \(|z+1| = 6\), and matches option (C).

Key Concepts

Inequalities

Inequalities are a fundamental concept in mathematics, including when dealing with complex numbers. An inequality relates two expressions using inequality signs such as \(<, >, \leq,\) or \(\geq\). In the context of complex numbers, inequalities often describe sets or regions within the complex plane. For example, the inequality \( |z+4| \leq 3 \) describes the set of all complex numbers \(z\) whose distance from the point \(-4\) is 3 or less. Here, the modulus \( |z+4| \) represents the distance of the complex number \(z = x + yi\) from \(-4, 0\) on the complex plane. This is an example of how inequalities can be used to define geometric shapes!
The inequality is essentially a property that restricts \(z\) to a specific circular region. When translating into an equation by squaring both sides \((x+4)^2 + y^2 \leq 9\), it confirms that the region is a circle centered at \(-4, 0\) with radius 3.

Circle in the complex plane

A circle in the complex plane is a collection of points that maintain a constant distance, known as the radius, from a fixed center. The circle defined by the inequality \( |z + 4| \leq 3 \) is crucial to solving this problem. Here, it means all points within or on the boundary of a defined circle.
In the complex plane, a circle centered at a point \((a, b)\) is represented by an equation such as \( |z - a - bi| = r \), where \(r\) is the radius. For \( |z+4| \leq 3 \), \(-4\) acts as the center (\