If a complex number z lies in the interior or on the boundary of a circle of radius 3 units and centre (−4,0), find the greatest and least values of ∣z+1∣.
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Step-by-Step Solution
Step 1: Understand the given condition
The problem states that the complex number z lies in the interior or on the boundary of a circle with radius 3 units and center (−4,0). In the complex plane, the distance between a complex number z and a complex number c is given by |z-c|. Therefore, the condition can be written as ∣z−(−4)∣≤3, which simplifies to ∣z+4∣≤3. This means z is a point within or on the circle centered at (−4,0) with radius 3.
Step 2: Interpret the expression to be maximized/minimized
We need to find the greatest and least values of ∣z+1∣. This expression represents the distance between the complex number z and the complex number −1 (which corresponds to the point (−1,0) in the complex plane).
Step 3: Visualize the geometry
Let the center of the given circle be C=(−4,0) and its radius be r=3. Let the point corresponding to the complex number −1 be P=(−1,0). We are looking for the maximum and minimum distances from any point z within or on the circle to the point P.
Step 4: Calculate the distance between the center and the point P
First, calculate the distance d between the center of the circle C(−4,0) and the point P(−1,0). This distance is ∣−4−(−1)∣=∣−3∣=3. Notice that this distance is equal to the radius of the circle.
Step 5: Determine the greatest value of ∣z+1∣
The greatest distance from point P to any point z on or inside the circle occurs when z is on the circle, on the line passing through P and C, and on the opposite side of C from P. This maximum distance is the sum of the distance d from P to C and the radius r. So, ∣z+1∣max=d+r=3+3=6.
Step 6: Determine the least value of ∣z+1∣
The least distance from point P to any point z on or inside the circle occurs when z is on the circle, on the line passing through P and C, and on the same side of C as P. This minimum distance is the absolute difference between the distance d from P to C and the radius r. Since d=r=3, the minimum distance is 3−3=0. This means the point P(−1,0) lies on the boundary of the circle, so z can be equal to −1, making ∣z+1∣=0.