The point on the x-axis which is equidistant from the points (7,6) and (-3, 4) is 

Options

A. (0, 3)
B. (3, 0)
C. (-3, 0)
D. (0, -3)

Answer

To find the point on the x-axis that is equidistant from the points (7, 6) and (-3, 4), we can follow these steps:

### Step 1: Define the Point on the X-axis
Let the point on the x-axis be \( C(x, 0) \), where \( x \) is the x-coordinate we need to find.

### Step 2: Use the Distance Formula
The distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
We will calculate the distances \( AC \) and \( BC \) where:
- \( A(7, 6) \)
- \( B(-3, 4) \)
- \( C(x, 0) \)

### Step 3: Calculate Distance AC
Using the distance formula for points \( A(7, 6) \) and \( C(x, 0) \):
\[ AC = \sqrt{(x - 7)^2 + (0 - 6)^2} = \sqrt{(x - 7)^2 + 36} \]

### Step 4: Calculate Distance BC
Using the distance formula for points \( B(-3, 4) \) and \( C(x, 0) \):
\[ BC = \sqrt{(x + 3)^2 + (0 - 4)^2} = \sqrt{(x + 3)^2 + 16} \]

### Step 5: Set Distances Equal
Since \( C \) is equidistant from \( A \) and \( B \), we set the distances equal:
\[ \sqrt{(x - 7)^2 + 36} = \sqrt{(x + 3)^2 + 16} \]

### Step 6: Square Both Sides
To eliminate the square roots, we square both sides:
\[ (x - 7)^2 + 36 = (x + 3)^2 + 16 \]

### Step 7: Expand Both Sides
Expanding both sides gives:
\[ (x^2 - 14x + 49) + 36 = (x^2 + 6x + 9) + 16 \]
This simplifies to:
\[ x^2 - 14x + 85 = x^2 + 6x + 25 \]

### Step 8: Simplify the Equation
Subtract \( x^2 \) from both sides:
\[ -14x + 85 = 6x + 25 \]
Now, bring all terms involving \( x \) to one side and constant terms to the other:
\[ -14x - 6x = 25 - 85 \]
This simplifies to:
\[ -20x = -60 \]

### Step 9: Solve for x
Dividing both sides by -20 gives:
\[ x = \frac{-60}{-20} = 3 \]

### Conclusion
The point on the x-axis that is equidistant from the points (7, 6) and (-3, 4) is \( (3, 0) \).

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