The equation $\left|z - 2\right| = \left|z - 2i\right|$ represents a straight line.

Explanation:

The given equation compares the distances of a complex number $z = x + yi$ from the two fixed points $2$ and $2i$ in the complex plane.

• The point $2$ corresponds to the complex number $2 + 0i$, which is located on the real axis at $(2, 0)$.
• The point $2i$ corresponds to the complex number $0 + 2i$, which is located on the imaginary axis at $(0, 2)$.

The equation $\left|z - 2\right| = \left|z - 2i\right|$ means that the distance from $z$ to $2$ is equal to the distance from $z$ to $2i$.

Geometric Interpretation:

The set of all points equidistant from two fixed points forms the perpendicular bisector of the line segment joining the two points.

• The midpoint of the segment joining $2$ and $2i$ is $\left(\frac{2 + 0}{2}, \frac{0 + 2}{2}\right) = (1, 1)$.
• The slope of the line segment joining $2$ and $2i$ is $-1$.

The perpendicular bisector of this segment will therefore be a line with a slope of $1$ passing through the midpoint $(1, 1)$. The equation of this line can be written as:

$y - 1 = 1(x - 1) \quad \text{or} \quad y = x$

So, the equation $\left|z - 2\right| = \left|z - 2i\right|$ represents the line $y = x$.

Conclusion:

The correct answer is straight line.

Do you have any further questions, or would you like more details on this topic? Here are some related questions you might consider:

1. How do you derive the equation of the perpendicular bisector in the complex plane?
2. What is the geometric interpretation of the modulus of a complex number?
3. How can we find the distance between two complex numbers geometrically?
4. What other geometric shapes can be represented by complex number equations?
5. What is the locus of points equidistant from a point and a line in the complex plane?

Tip: When working with complex numbers, visualizing the problem on the complex plane can often simplify the understanding of geometric properties.