If `z_1, z_2 in C , z_1^2 in R , z_1(z_1^2-3z_2^2)=2` and `z_2(3z_1^2-z_2^2)=11`, then the value of `z_1^2+z_2^2` is

Options

A. 10
B. 12
C. 5
D. 8

Explanation

To solve the problem, we start with the given equations:

1. \( z_1(z_1^2 - 3z_2^2) = 2 \) (Equation 1)
2. \( z_2(3z_1^2 - z_2^2) = 11 \) (Equation 2)

### Step 1: Rewrite the equations
We can rewrite the equations as follows:

From Equation 1:
\[ z_1^3 - 3z_1z_2^2 = 2 \]

From Equation 2:
\[ 3z_1^2z_2 - z_2^3 = 11 \]

### Step 2: Introduce complex numbers
Let \( z_1 = a + bi \) and \( z_2 = c + di \), where \( a, b, c, d \) are real numbers. Since \( z_1^2 \) is real, \( b = 0 \) (because the imaginary part must vanish). Therefore, \( z_1 = a \) (a real number).

### Step 3: Substitute \( z_1 \) into the equations
Substituting \( z_1 = a \) into the equations gives:

1. \( a(a^2 - 3z_2^2) = 2 \)
2. \( z_2(3a^2 - z_2^2) = 11 \)

Let \( z_2^2 = x \), then we can rewrite the equations as:

1. \( a(a^2 - 3x) = 2 \) (Equation 3)
2. \( z_2(3a^2 - x) = 11 \) (Equation 4)

### Step 4: Solve for \( z_2 \)
From Equation 4, we can express \( z_2 \) in terms of \( x \):
\[ z_2 = \frac{11}{3a^2 - x} \]

### Step 5: Substitute \( z_2 \) back into Equation 3
Substituting \( z_2 \) into Equation 3 gives:
\[ a(a^2 - 3\left(\frac{11}{3a^2 - x}\right)^2) = 2 \]

### Step 6: Simplify and solve for \( a \) and \( x \)
This equation can be complicated, so we can alternatively multiply the two original equations directly as follows:

### Step 7: Multiply the two equations
Multiply Equation 1 by \( i \) and Equation 2 by \( -i \):
\[ z_1(z_1^2 - 3z_2^2) + i z_2(3z_1^2 - z_2^2) = 2 + 11i \]
\[ z_1(z_1^2 - 3z_2^2) - i z_2(3z_1^2 - z_2^2) = 2 - 11i \]

### Step 8: Form a cube
This can be recognized as:
\[ (z_1 + iz_2)^3 = 2 + 11i \]
\[ (z_1 - iz_2)^3 = 2 - 11i \]

### Step 9: Multiply the two results
Now, multiply these two equations:
\[ (z_1^2 + z_2^2)^3 = (2 + 11i)(2 - 11i) = 4 + 121 = 125 \]

### Step 10: Take the cube root
Taking the cube root gives:
\[ z_1^2 + z_2^2 = 5 \]

### Conclusion
Thus, the value of \( z_1^2 + z_2^2 \) is \( \boxed{5} \).