Find the point on the x-axis which is equidistant from `(2,\- 5)\ and\ (-2,\ 9)`

Options

A. `(-7,0)`
B. `(7,0)`
C. `(-2,0)`
D. `(-9,0)`

Explanation

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

### Step 1: Define the Point on the X-axis
Since we are looking for a point on the x-axis, we can denote this point as \( (x, 0) \).

### Step 2: Use the Distance Formula
We need to find the distances from the point \( (x, 0) \) to the two given points. The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

### Step 3: Calculate Distance to the First Point
The distance \( AO \) from \( (x, 0) \) to \( (2, -5) \) is:

\[ AO = \sqrt{(x - 2)^2 + (0 - (-5))^2} = \sqrt{(x - 2)^2 + 5^2} = \sqrt{(x - 2)^2 + 25} \]

### Step 4: Calculate Distance to the Second Point
The distance \( OB \) from \( (x, 0) \) to \( (-2, 9) \) is:

\[ OB = \sqrt{(x - (-2))^2 + (0 - 9)^2} = \sqrt{(x + 2)^2 + 9^2} = \sqrt{(x + 2)^2 + 81} \]

### Step 5: Set the Distances Equal
Since the point \( (x, 0) \) is equidistant from both points, we set the distances equal:

\[ \sqrt{(x - 2)^2 + 25} = \sqrt{(x + 2)^2 + 81} \]

### Step 6: Square Both Sides
To eliminate the square roots, we square both sides:

\[ (x - 2)^2 + 25 = (x + 2)^2 + 81 \]

### Step 7: Expand Both Sides
Expanding both sides gives:

\[ (x^2 - 4x + 4) + 25 = (x^2 + 4x + 4) + 81 \]

This simplifies to:

\[ x^2 - 4x + 29 = x^2 + 4x + 85 \]

### Step 8: Simplify the Equation
Now, we can simplify the equation by subtracting \( x^2 \) from both sides:

\[ -4x + 29 = 4x + 85 \]

### Step 9: Combine Like Terms
Rearranging gives:

\[ -4x - 4x = 85 - 29 \]

This simplifies to:

\[ -8x = 56 \]

### Step 10: Solve for x
Dividing both sides by -8 gives:

\[ x = -7 \]

### Step 11: Write the Final Point
The point on the x-axis is \( (-7, 0) \).

### Final Answer
The required point is \( (-7, 0) \).

---