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

Answer

To find a 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 \( R(x, 0) \), where \( x \) is the x-coordinate we need to find.

### Step 2: Calculate the Distance from Point R to Points P and Q
We need to calculate the distances from point \( R \) to the two given points \( P(7, 6) \) and \( Q(-3, 4) \).

1. **Distance \( RP \)** from \( R(x, 0) \) to \( P(7, 6) \):
\[ RP = \sqrt{(x - 7)^2 + (0 - 6)^2} = \sqrt{(x - 7)^2 + 36} \]

2. **Distance \( RQ \)** from \( R(x, 0) \) to \( Q(-3, 4) \):
\[ RQ = \sqrt{(x + 3)^2 + (0 - 4)^2} = \sqrt{(x + 3)^2 + 16} \]

### Step 3: Set the Distances Equal
Since point \( R \) is equidistant from points \( P \) and \( Q \), we set the distances equal to each other:
\[ \sqrt{(x - 7)^2 + 36} = \sqrt{(x + 3)^2 + 16} \]

### Step 4: Square Both Sides to Eliminate the Square Roots
Squaring both sides gives:
\[ (x - 7)^2 + 36 = (x + 3)^2 + 16 \]

### Step 5: Expand Both Sides
1. Expanding the left side:
\[ (x - 7)^2 = x^2 - 14x + 49 \implies x^2 - 14x + 49 + 36 = x^2 - 14x + 85 \]

2. Expanding the right side:
\[ (x + 3)^2 = x^2 + 6x + 9 \implies x^2 + 6x + 9 + 16 = x^2 + 6x + 25 \]

### Step 6: Set the Expanded Equations Equal
Now we have:
\[ x^2 - 14x + 85 = x^2 + 6x + 25 \]

### Step 7: Simplify the Equation
Subtract \( x^2 \) from both sides:
\[ -14x + 85 = 6x + 25 \]

Combine like terms:
\[ -14x - 6x = 25 - 85 \]
\[ -20x = -60 \]

### Step 8: Solve for x
Dividing both sides by -20:
\[ x = \frac{-60}{-20} = 3 \]

### Step 9: Write the Final Point
The point \( R \) on the x-axis is:
\[ R(3, 0) \]

### 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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