What will be the point on the x- axis which is eduidistant from the points ( 7, 6) and ( - 3, 6)

Options

A. (6,0)
B. (-2,0)
C. (4,0)
D. (3,0)

Answer

To find the point on the x-axis that is equidistant from the points (7, 6) and (-3, 6), we can follow these steps:

### Step 1: Identify the coordinates of the points
The two points given are:
- Point A: (7, 6)
- Point B: (-3, 6)

### Step 2: Understand the requirement
We need to find a point on the x-axis, which means its y-coordinate will be 0. Let's denote this point as (x, 0).

### Step 3: Use the distance formula
The distance from point (x, 0) to point A (7, 6) can be calculated using the distance formula:
\[ d_A = \sqrt{(x - 7)^2 + (0 - 6)^2} \]
This simplifies to:
\[ d_A = \sqrt{(x - 7)^2 + 36} \]

The distance from point (x, 0) to point B (-3, 6) is:
\[ d_B = \sqrt{(x + 3)^2 + (0 - 6)^2} \]
This simplifies to:
\[ d_B = \sqrt{(x + 3)^2 + 36} \]

### Step 4: Set the distances equal
Since we want the point (x, 0) to be equidistant from points A and B, we set the distances equal to each other:
\[ \sqrt{(x - 7)^2 + 36} = \sqrt{(x + 3)^2 + 36} \]

### Step 5: Square both sides to eliminate the square roots
Squaring both sides gives us:
\[ (x - 7)^2 + 36 = (x + 3)^2 + 36 \]

### Step 6: Simplify the equation
We can cancel out the 36 from both sides:
\[ (x - 7)^2 = (x + 3)^2 \]

### Step 7: Expand both sides
Expanding both sides results in:
\[ x^2 - 14x + 49 = x^2 + 6x + 9 \]

### Step 8: Rearrange the equation
Subtract \(x^2\) from both sides:
\[ -14x + 49 = 6x + 9 \]
Now, move all terms involving x to one side and constant terms to the other:
\[ -14x - 6x = 9 - 49 \]
This simplifies to:
\[ -20x = -40 \]

### Step 9: Solve for x
Dividing both sides by -20 gives:
\[ x = 2 \]

### Step 10: Write the final point
Since we are looking for the point on the x-axis, the coordinates are:
\[ (2, 0) \]

### Conclusion
The point on the x-axis that is equidistant from the points (7, 6) and (-3, 6) is **(2, 0)**.

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