Let `z_1=3` and `z_2=7` represent two points A and B respectively on complex plane . Let the curve `C_1` be the locus of pint P(z) satisfying `|z-z_1|^2 + |z-z_2|^2 =10` and the curve `C_2` be the locus of point P(z) satisfying `|z-z_1|^2 + |z-z_2|^2 =16`
Least distance between curves `C_1` and `C_2` is :

Options

A. A) 4
B. B) 3
C. C) 2
D. D) 1

Answer

To solve the problem, we need to analyze the given curves \( C_1 \) and \( C_2 \) based on the conditions provided.

### Step-by-Step Solution:

1. **Identify the Points**:
Given:
\[ z_1 = 3 \quad \text{and} \quad z_2 = 7 \]
These represent points \( A(3, 0) \) and \( B(7, 0) \) on the complex plane.

2. **Write the Equations for the Curves**:
The curves are defined by:
\[ C_1: |z - z_1|^2 + |z - z_2|^2 = 10 \]
\[ C_2: |z - z_1|^2 + |z - z_2|^2 = 16 \]

3. **Substitute the Values**:
Substitute \( z_1 \) and \( z_2 \) into the equations:
\[ C_1: |z - 3|^2 + |z - 7|^2 = 10 \]
\[ C_2: |z - 3|^2 + |z - 7|^2 = 16 \]

4. **Express \( z \) in Terms of \( x \) and \( y \)**:
Let \( z = x + iy \). Then:
\[ |z - 3|^2 = (x - 3)^2 + y^2 \]
\[ |z - 7|^2 = (x - 7)^2 + y^2 \]

5. **Substitute into the Curves**:
For \( C_1 \):
\[ (x - 3)^2 + y^2 + (x - 7)^2 + y^2 = 10 \]
Simplifying:
\[ (x^2 - 6x + 9) + y^2 + (x^2 - 14x + 49) + y^2 = 10 \]
\[ 2x^2 - 20x + 58 + 2y^2 = 10 \]
\[ 2x^2 + 2y^2 - 20x + 48 = 0 \]
Dividing by 2:
\[ x^2 + y^2 - 10x + 24 = 0 \]
Completing the square:
\[ (x - 5)^2 + y^2 = 1 \]

For \( C_2 \):
\[ (x - 3)^2 + y^2 + (x - 7)^2 + y^2 = 16 \]
Following similar steps:
\[ 2x^2 - 20x + 58 + 2y^2 = 16 \]
\[ 2x^2 + 2y^2 - 20x + 42 = 0 \]
Dividing by 2:
\[ x^2 + y^2 - 10x + 21 = 0 \]
Completing the square:
\[ (x - 5)^2 + y^2 = 4 \]

6. **Identify the Circles**:
The equations represent circles:
- \( C_1: (x - 5)^2 + y^2 = 1 \) (radius = 1)
- \( C_2: (x - 5)^2 + y^2 = 4 \) (radius = 2)

7. **Calculate the Least Distance**:
Since both circles are concentric (same center at \( (5, 0) \)), the least distance between them is the difference of their radii:
\[ \text{Distance} = \text{Radius of } C_2 - \text{Radius of } C_1 = 2 - 1 = 1 \]

### Conclusion:
The least distance between curves \( C_1 \) and \( C_2 \) is \( 1 \).