Locus of the point z satisfying the equation ` |zi-i|+ |z-i| =2` is
(1) A line segment (2) A circle
(3) An eplipse (4) A pair of straight line

Answer

To find the locus of the point \( z \) satisfying the equation \( |zi - i| + |z - i| = 2 \), we can follow these steps:

### Step 1: Rewrite the equation
The equation can be rewritten as:
\[ |z(i - 1)| + |z - i| = 2 \]

### Step 2: Interpret the terms
Here, \( |z(i - 1)| \) represents the distance from the point \( z \) to the point \( i \) (which is \( (0, 1) \) in the complex plane) scaled by the factor \( |i - 1| \). The term \( |z - i| \) represents the distance from the point \( z \) to the point \( i \).

### Step 3: Identify the points involved
The points involved in the equation are \( i \) (which is \( (0, 1) \)) and \( 1 \) (which is \( (1, 0) \)). The distance \( |z - i| \) is the distance from \( z \) to the point \( i \), and \( |zi - i| \) can be interpreted as the distance from \( z \) to the point \( 1 \).

### Step 4: Analyze the equation
The equation \( |z - i| + |z - 1| = 2 \) describes a geometric locus. This is the definition of an ellipse where the sum of the distances from any point \( z \) on the ellipse to the two foci (the points \( i \) and \( 1 \)) is constant.

### Step 5: Determine the nature of the locus
Since the sum of the distances is constant (equal to 2), and the distance between the two points \( i \) and \( 1 \) is less than 2, the locus is indeed an ellipse.

### Final Conclusion
Thus, the locus of the point \( z \) satisfying the equation \( |zi - i| + |z - i| = 2 \) is an ellipse.

### Answer
(3) An ellipse

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