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

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 represented as \( P(a, 0) \), where \( a \) is the x-coordinate we need to find.

### Step 2: Calculate the distance from P to (7, 6)
Using the distance formula, the distance \( d_1 \) from point \( P(a, 0) \) to point \( (7, 6) \) is given by:
\[ d_1 = \sqrt{(a - 7)^2 + (0 - 6)^2} \]
This simplifies to:
\[ d_1 = \sqrt{(a - 7)^2 + 36} \]

### Step 3: Calculate the distance from P to (-3, 4)
Similarly, the distance \( d_2 \) from point \( P(a, 0) \) to point \( (-3, 4) \) is:
\[ d_2 = \sqrt{(a + 3)^2 + (0 - 4)^2} \]
This simplifies to:
\[ d_2 = \sqrt{(a + 3)^2 + 16} \]

### Step 4: Set the distances equal
Since point \( P(a, 0) \) is equidistant from both points, we set the distances equal:
\[ \sqrt{(a - 7)^2 + 36} = \sqrt{(a + 3)^2 + 16} \]

### Step 5: Square both sides to eliminate the square roots
Squaring both sides gives:
\[ (a - 7)^2 + 36 = (a + 3)^2 + 16 \]

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

### Step 7: Simplify the equation
Subtract \( a^2 \) from both sides:
\[ -14a + 85 = 6a + 25 \]
Now, move all terms involving \( a \) to one side and constant terms to the other side:
\[ -14a - 6a = 25 - 85 \]
This simplifies to:
\[ -20a = -60 \]

### Step 8: Solve for \( a \)
Dividing both sides by -20 gives:
\[ a = 3 \]

### 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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