Problem 205 - A Complete Resource Book in Mathematics for JEE Main 2017

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

(C) 6

Step 1: Understand the inequality

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

Step 2: Analyze the expression \\( |z+1| \\)

We need to find the maximum value of \(|z+1|\). This represents the distance from any point \(z\) to \(-1\) on the complex plane.

Step 3: Visualize the loci of points

The center of the circle defined by \(|z+4| \leq 3\) is at \(-4\). We need to find the maximum distance from this circle to the point \(-1\). The maximum value will be at the boundary of the circle away from \(-1\).

Step 4: Calculate the maximum distance

The straightforward approach is to calculate the distance from \(-4\) to \(-1\) and add the radius 3. The distance from \(-4\) to \(-1\) is \[|-4 - (-1)| = |-4 + 1| = 3.\]The maximum distance would be \(|z+1| = 3 + 3 = 6\) where the periphery of the circle is away from \(-1\).

Step 5: Conclude with the answer

Hence, the maximum value of \(|z+1|\) is the total distance from \(-1\) to the farthest point on the given circle.

Key Concepts

Geometric Interpretation

Understanding complex numbers from a geometric perspective can really clarify their properties and relationships. Imagine an ordinary number line; each point on this line corresponds to a real number. When dealing with complex numbers, instead of a line, we use a plane, commonly referred to as the "complex plane." In this plane:

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

Every complex number can be thought of as a point or a vector on this plane.

Thus, the inequality \(|z+4| \leq 3\) can be interpreted geometrically as a circle. This circle has a center at the point \(-4\) on the real axis and a radius of 3. Understanding these geometric formations helps in visualizing how complex numbers behave, allowing us to solve problems more intuitively.

It sometimes helps to draw these circles on graph paper and label the center and radius properly. Such practices deepen understanding and make complex operations less abstract.

Maximum Distance

In the problem above, we're interested in finding the maximum distance to a specific point on the complex plane from a given set of points. This is determined by a circle defined by \(|z+4| \leq 3\).

The expression \(|z+1|\) represents the distance from any point \(`z`\) on this circle to another point \(-1\). To find the maximum distance, consider the total path stretched from \(-4\) (the center of the initial circle) to \(-1\) (our point of reference).

Here's how you determine it:

• Start by calculating the distance from \(-4\) to \(-1\) on the real axis. This gives 3, just like a straight line segment between these points.
• Then, add the radius of our circle, which is also 3.

So, essentially, our total maximum distance \( \(|z+1|\) \) becomes \(3 + 3 = 6\). This shows how geometric interpretation blends beautifully with algebra to solve complex number problems efficiently.

Inequalities in Complex Numbers

When approaching inequalities involving complex numbers, the magnitude of a complex number, \(|z|\), is often at the forefront. This magnitude is essentially the distance of a complex number from the origin in the complex plane.

Inequalities in this realm can describe regions within the plane, like circles or even bounded areas when combined. For example, \( |z + 4| \leq 3 \) implies that the point represented by \( z \) is at most 3 units away from \(-4\).

When solving these inequalities:

• Consider how each term in \( z = x + yi \) affects the distance measured.
• Use geometric shapes, like circles, to visualize where \( z \) can be on the plane.

These insights provide powerful tools in determining complex relationships and solutions. In practice, visualizing inequalities aids greatly in cutting through complex algebra to isolate the solution.