3 (a) Show graphically the geometric image of the complex number: \( |z+i| \geq 2 \)

To graphically represent the inequality \( |z + i| \geq 2 \) in the complex plane, let's break it down step by step. ### 1. Understanding the Expression Let \( z \) be a complex number represented as \( z = x + iy \), where: - \( x \) is the real part. - \( y \) is the imaginary part. The expression \( |z + i| \) represents the **distance** between the complex number \( z \) and the point \( -i \) in the complex plane. ### 2. Rewriting the Inequality Express \( |z + i| \) in terms of \( x \) and \( y \): \[ |z + i| = |x + iy + i| = |x + i(y + 1)| = \sqrt{x^2 + (y + 1)^2} \] So, the inequality \( |z + i| \geq 2 \) becomes: \[ \sqrt{x^2 + (y + 1)^2} \geq 2 \] ### 3. Squaring Both Sides To simplify, square both sides of the inequality: \[ x^2 + (y + 1)^2 \geq 4 \] ### 4. Geometric Interpretation This inequality represents all the points \( (x, y) \) in the complex plane that are **at least a distance of 2 units** away from the point \( (0, -1) \). Here's what this looks like: - **Center of the Circle**: \( (0, -1) \) - **Radius of the Circle**: \( 2 \) ### 5. Graphical Representation - **Draw the Complex Plane**: Plot the real axis (horizontal) and the imaginary axis (vertical). - **Locate the Center**: Mark the point \( (0, -1) \) on the imaginary axis. - **Draw the Circle**: With a radius of 2 units around \( (0, -1) \), draw a circle. - **Shade the Region**: Since the inequality is \( \geq 2 \), shade the entire **exterior** of the circle, including the boundary. ### 6. Visual Summary ``` Imaginary Axis (y) | | Exterior Region (|z + i| ≥ 2) | ___________ | / \ | / \ | / \ |_____/_________________\_____ Real Axis (x) (0, -1) ``` - **Circle**: Centered at \( (0, -1) \) with radius 2. - **Shaded Area**: All points outside and on the circle. ### 7. Interpretation - **Points Inside the Circle**: These satisfy \( |z + i| < 2 \) and are **not** included in the region defined by the inequality. - **Points On and Outside the Circle**: These satisfy \( |z + i| \geq 2 \) and are included in the region. ### 8. Conclusion The geometric image of \( |z + i| \geq 2 \) in the complex plane is the **set of all points lying on or outside a circle** centered at \( (0, -1) \) with a radius of 2 units.