Find the point on the X - axis which is equidistant from (2,-5) and (-2,9).

Answer

To find the point on the X-axis that is equidistant from the points (2, -5) and (-2, 9), we can follow these steps:

### Step 1: Understand the Problem
We need to find a point C on the X-axis. Since C is on the X-axis, its coordinates can be represented as (x, 0).

### Step 2: Set Up the Distance Equations
The distance from point A (2, -5) to point C (x, 0) must be equal to the distance from point B (-2, 9) to point C (x, 0).

Using the distance formula:
- Distance AC = √[(x - 2)² + (0 - (-5))²] = √[(x - 2)² + 5²]
- Distance BC = √[(x - (-2))² + (0 - 9)²] = √[(x + 2)² + 9²]

Since these distances are equal, we can set them equal to each other:
\[ \sqrt{(x - 2)² + 25} = \sqrt{(x + 2)² + 81} \]

### Step 3: Square Both Sides
To eliminate the square roots, we square both sides:
\[ (x - 2)² + 25 = (x + 2)² + 81 \]

### Step 4: Expand Both Sides
Now, we expand both sides of the equation:
- Left side: (x - 2)² = x² - 4x + 4
- Right side: (x + 2)² = x² + 4x + 4

So, we have:
\[ x² - 4x + 4 + 25 = x² + 4x + 4 + 81 \]
This simplifies to:
\[ x² - 4x + 29 = x² + 4x + 85 \]

### Step 5: Simplify the Equation
Now, we can cancel x² from both sides:
\[ -4x + 29 = 4x + 85 \]

### Step 6: Rearrange the Equation
Rearranging gives:
\[ -4x - 4x = 85 - 29 \]
\[ -8x = 56 \]

### Step 7: Solve for x
Dividing both sides by -8 gives:
\[ x = -7 \]

### Step 8: Write the Coordinates of Point C
Since C is on the X-axis, its coordinates are:
\[ C = (-7, 0) \]

### Final Answer
The point on the X-axis which is equidistant from (2, -5) and (-2, 9) is:
\[ \boxed{(-7, 0)} \]

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