Find the minimum value of the expression `E= |z|^2+ |z-3|^2 + |z- 6i|^2` (where `z=x+iy, x,y in R`)

Explanation

To find the minimum value of the expression \( E = |z|^2 + |z - 3|^2 + |z - 6i|^2 \), where \( z = x + iy \) (with \( x, y \in \mathbb{R} \)), we will follow these steps:

### Step 1: Substitute \( z \)
We start by substituting \( z = x + iy \) into the expression. Thus, we have:
\[ |z|^2 = |x + iy|^2 = x^2 + y^2 \]

### Step 2: Calculate \( |z - 3|^2 \)
Next, we calculate \( |z - 3|^2 \):
\[ |z - 3|^2 = |(x - 3) + iy|^2 = (x - 3)^2 + y^2 \]

### Step 3: Calculate \( |z - 6i|^2 \)
Now, we calculate \( |z - 6i|^2 \):
\[ |z - 6i|^2 = |x + (y - 6)i|^2 = x^2 + (y - 6)^2 \]

### Step 4: Combine the expressions
Now we can combine all the expressions we have calculated for \( E \):
\[ E = |z|^2 + |z - 3|^2 + |z - 6i|^2 \]
Substituting the values we found:
\[ E = (x^2 + y^2) + ((x - 3)^2 + y^2) + (x^2 + (y - 6)^2) \]

### Step 5: Expand the expression
Now, we expand the expression:
\[ E = x^2 + y^2 + (x^2 - 6x + 9 + y^2) + (x^2 + y^2 - 12y + 36) \]
Combining like terms:
\[ E = 3x^2 + 3y^2 - 6x - 12y + 45 \]

### Step 6: Rearrange the expression
We can rearrange the expression to make it easier to find the minimum:
\[ E = 3(x^2 - 2x) + 3(y^2 - 4y) + 45 \]

### Step 7: Complete the square
Now we complete the square for both \( x \) and \( y \):
\[ E = 3((x - 1)^2 - 1) + 3((y - 2)^2 - 4) + 45 \]
This simplifies to:
\[ E = 3(x - 1)^2 + 3(y - 2)^2 + 45 - 3 - 12 \]
\[ E = 3(x - 1)^2 + 3(y - 2)^2 + 30 \]

### Step 8: Find the minimum value
The minimum value of \( E \) occurs when both squares are zero, i.e., when \( x - 1 = 0 \) and \( y - 2 = 0 \). Thus:
\[ x = 1, \quad y = 2 \]
Substituting these values back into the expression for \( E \):
\[ E_{\text{min}} = 3(0) + 3(0) + 30 = 30 \]

### Final Answer
The minimum value of the expression \( E \) is \( \boxed{30} \).

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