Let ` z_(1) and z_(2) ` be two given complex numbers. The locus of z such that
`{:("Column -I", " Column -II"),( "(A) " |z-z_(1)|+|z-z_(2)| = " constant =k, where " k ne|z_(1)-z_(2)|, " (p) Circle with " z_(1) and z_(2) " as the vertices of diameter"),("(B)" |z-z_(1)|- |z-z_(2)|= " k where " k ne |z_(1)-z_(2)| ," (q) Circle "),("(C)"arg((z-z_(1))/(z-z_(2)))=+- pi/2 , " (r) Hyperbola "),("(D) If "omega" lies on " |omega| = 1 " then " , " (s) Ellipse"):}`

Answer

To solve the problem, we need to analyze each of the given conditions for the complex number \( z \) and determine the corresponding geometric shapes.

### Step-by-Step Solution:

1. **Option (A):**
\[ |z - z_1| + |z - z_2| = k \quad \text{where } k \neq |z_1 - z_2| \]
- This equation represents the sum of the distances from the point \( z \) to the fixed points \( z_1 \) and \( z_2 \).
- The locus of points where the sum of distances to two fixed points is constant (and not equal to the distance between the two points) is an **ellipse**.
- **Match:** (A) → (s) Ellipse

2. **Option (B):**
\[ |z - z_1| - |z - z_2| = k \quad \text{where } k \neq |z_1 - z_2| \]
- This equation represents the difference of distances from the point \( z \) to the fixed points \( z_1 \) and \( z_2 \).
- The locus of points where the difference of distances to two fixed points is constant (and not equal to the distance between the two points) is a **hyperbola**.
- **Match:** (B) → (r) Hyperbola

3. **Option (C):**
\[ \arg\left(\frac{z - z_1}{z - z_2}\right) = \pm \frac{\pi}{2} \]
- This condition implies that the angle formed by the line segments from \( z \) to \( z_1 \) and from \( z \) to \( z_2 \) is \( 90^\circ \).
- This means that the points \( z \) lie on a circle with \( z_1 \) and \( z_2 \) as the endpoints of a diameter.
- **Match:** (C) → (p) Circle

4. **Option (D):**
\[ \text{If } \omega \text{ lies on } | \omega | = 1 \]
- The condition \( |\omega| = 1 \) indicates that \( \omega \) lies on the unit circle in the complex plane.
- This does not directly correspond to any of the previous conditions but is a standard condition for a circle.
- Since there is no specific geometric shape mentioned for this option, we can conclude that it does not match with any of the other options.
- **Match:** (D) → (q) Circle (as it relates to the unit circle)

### Final Matches:
- (A) → (s) Ellipse
- (B) → (r) Hyperbola
- (C) → (p) Circle
- (D) → (q) Circle