Let `z_(0)` be the circumcenter of an equilateral triangle whose affixes are `z_(1),z_(2),z_(3)`.
Statement-1 : `z_(1)^(2)+z_(2)^(2)+z_(3)^(2)=3z_(0)^(2)`
Statement-2: `z_(1)^(2)+z_(2)^(2)+z_(3)^(2)=2(z_(1)z_(2)+z_(2)z_(3)+z_(3)z_(1))`

Options

A. Statement-1 is True, Statement-2 is True: Statement-2 is a correct exp,anation for statement-1.
B. Statement-1 is true, statement -2 is true, Statement-2 is not a correct explanation for statement-1.
C. Statement-1 is True, statement-2 is false,
D. statement-1 is False, Statement-2 is true.

Answer

To solve the problem, we need to analyze the two statements regarding the circumcenter \( z_0 \) of an equilateral triangle with affixes \( z_1, z_2, z_3 \).

### Step 1: Understanding the circumcenter of an equilateral triangle
The circumcenter \( z_0 \) of an equilateral triangle is the same as its centroid. Therefore, we can express \( z_0 \) as:
\[ z_0 = \frac{z_1 + z_2 + z_3}{3} \]

### Step 2: Analyzing Statement 1
We need to verify the statement:
\[ z_1^2 + z_2^2 + z_3^2 = 3z_0^2 \]
Substituting \( z_0 \):
\[ z_0^2 = \left(\frac{z_1 + z_2 + z_3}{3}\right)^2 = \frac{(z_1 + z_2 + z_3)^2}{9} \]
Using the identity \( (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) \):
\[ (z_1 + z_2 + z_3)^2 = z_1^2 + z_2^2 + z_3^2 + 2(z_1z_2 + z_2z_3 + z_3z_1) \]
Substituting this back:
\[ z_0^2 = \frac{z_1^2 + z_2^2 + z_3^2 + 2(z_1z_2 + z_2z_3 + z_3z_1)}{9} \]
Thus,
\[ 3z_0^2 = \frac{1}{3}(z_1^2 + z_2^2 + z_3^2 + 2(z_1z_2 + z_2z_3 + z_3z_1)) \]
Multiplying through by 3 gives:
\[ 9z_0^2 = z_1^2 + z_2^2 + z_3^2 + 2(z_1z_2 + z_2z_3 + z_3z_1) \]
This simplifies to:
\[ z_1^2 + z_2^2 + z_3^2 = 3z_0^2 \]
So, **Statement 1 is true**.

### Step 3: Analyzing Statement 2
We need to verify the statement:
\[ z_1^2 + z_2^2 + z_3^2 = 2(z_1z_2 + z_2z_3 + z_3z_1) \]
From our previous derivation, we found:
\[ z_1^2 + z_2^2 + z_3^2 = z_1z_2 + z_2z_3 + z_3z_1 \]
This implies that:
\[ z_1^2 + z_2^2 + z_3^2 \neq 2(z_1z_2 + z_2z_3 + z_3z_1) \]
Thus, **Statement 2 is false**.

### Conclusion
- **Statement 1 is true**: \( z_1^2 + z_2^2 + z_3^2 = 3z_0^2 \)
- **Statement 2 is false**: \( z_1^2 + z_2^2 + z_3^2 \neq 2(z_1z_2 + z_2z_3 + z_3z_1) \)