Find the point in the XY plane which is equidistant from the points ` (2,0,3) ,(0,3,2)` and (0,0,1)

Answer

To find the point in the XY plane that is equidistant from the points \( A(2, 0, 3) \), \( B(0, 3, 2) \), and \( C(0, 0, 1) \), we can follow these steps:

### Step 1: Define the point in the XY plane
Let \( P(x, y, 0) \) be the point in the XY plane that we are trying to find.

### Step 2: Set up the distance equations
Since \( P \) is equidistant from points \( A \), \( B \), and \( C \), we can set up the following equations based on the distances:

1. The distance from \( P \) to \( A \) is equal to the distance from \( P \) to \( B \):
\[ PA = PB \]

2. The distance from \( P \) to \( B \) is equal to the distance from \( P \) to \( C \):
\[ PB = PC \]

### Step 3: Calculate \( PA^2 = PB^2 \)
Using the distance formula, we can express \( PA^2 \) and \( PB^2 \):

\[ PA^2 = (x - 2)^2 + (y - 0)^2 + (0 - 3)^2 \]
\[ PB^2 = (x - 0)^2 + (y - 3)^2 + (0 - 2)^2 \]

Setting these equal gives:
\[ (x - 2)^2 + y^2 + 9 = x^2 + (y - 3)^2 + 4 \]

### Step 4: Expand and simplify
Expanding both sides:
\[ (x^2 - 4x + 4) + y^2 + 9 = x^2 + (y^2 - 6y + 9) + 4 \]

This simplifies to:
\[ -4x + 13 = -6y + 13 \]

Removing 13 from both sides:
\[ -4x = -6y \]

Rearranging gives:
\[ 2x - 3y = 0 \quad \text{(Equation 1)} \]

### Step 5: Calculate \( PB^2 = PC^2 \)
Now, we calculate \( PB^2 \) and \( PC^2 \):

\[ PB^2 = (x - 0)^2 + (y - 3)^2 + (0 - 2)^2 \]
\[ PC^2 = (x - 0)^2 + (y - 0)^2 + (0 - 1)^2 \]

Setting these equal gives:
\[ x^2 + (y - 3)^2 + 4 = x^2 + y^2 + 1 \]

### Step 6: Expand and simplify
Expanding both sides:
\[ x^2 + (y^2 - 6y + 9) + 4 = x^2 + y^2 + 1 \]

This simplifies to:
\[ -6y + 13 = 1 \]

Removing 1 from both sides:
\[ -6y = -12 \]

Thus, we find:
\[ y = 2 \]

### Step 7: Substitute \( y \) back into Equation 1
Now, substitute \( y = 2 \) back into Equation 1:
\[ 2x - 3(2) = 0 \]
\[ 2x - 6 = 0 \]
\[ 2x = 6 \implies x = 3 \]

### Step 8: Conclusion
The point \( P \) in the XY plane is:
\[ P(3, 2, 0) \]

### Final Answer
The required point is \( (3, 2, 0) \).

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