Find a point on Z-axis which is equidistant from the points (1,5,7) and (5,1,-4).

Explanation

To find a point on the Z-axis that is equidistant from the points (1, 5, 7) and (5, 1, -4), we can follow these steps:

### Step 1: Define the Point on the Z-axis
Let the point on the Z-axis be \( R(0, 0, z) \), where \( z \) is the unknown coordinate we need to find.

**Hint:** Remember that any point on the Z-axis has its x and y coordinates equal to 0.

### Step 2: Use the Distance Formula
The distance between two points \( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \) in 3D space is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

### Step 3: Calculate the Distances
We need to find the distances \( RA \) and \( RB \):
- Distance \( RA \) from point \( R(0, 0, z) \) to point \( A(1, 5, 7) \):
\[ RA = \sqrt{(0 - 1)^2 + (0 - 5)^2 + (z - 7)^2} = \sqrt{1 + 25 + (z - 7)^2} = \sqrt{26 + (z - 7)^2} \]

- Distance \( RB \) from point \( R(0, 0, z) \) to point \( B(5, 1, -4) \):
\[ RB = \sqrt{(0 - 5)^2 + (0 - 1)^2 + (z + 4)^2} = \sqrt{25 + 1 + (z + 4)^2} = \sqrt{26 + (z + 4)^2} \]

### Step 4: Set the Distances Equal
Since point \( R \) is equidistant from points \( A \) and \( B \), we can set the distances equal:
\[ RA = RB \]
Squaring both sides to eliminate the square roots:
\[ 26 + (z - 7)^2 = 26 + (z + 4)^2 \]

### Step 5: Simplify the Equation
We can simplify the equation by canceling \( 26 \) from both sides:
\[ (z - 7)^2 = (z + 4)^2 \]

### Step 6: Expand Both Sides
Expanding both sides gives:
\[ z^2 - 14z + 49 = z^2 + 8z + 16 \]

### Step 7: Rearrange the Equation
Subtract \( z^2 \) from both sides:
\[ -14z + 49 = 8z + 16 \]
Rearranging gives:
\[ 49 - 16 = 8z + 14z \]
\[ 33 = 22z \]

### Step 8: Solve for \( z \)
Dividing both sides by \( 22 \):
\[ z = \frac{33}{22} = \frac{3}{2} \]

### Step 9: Write the Final Point
Thus, the point \( R \) on the Z-axis is:
\[ R(0, 0, \frac{3}{2}) \]

### Summary
The point on the Z-axis that is equidistant from the points (1, 5, 7) and (5, 1, -4) is \( R(0, 0, \frac{3}{2}) \).

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