Find the coordinates of a point which is equidistant from the two points (-4, 0) and (4, 0). How many of such points are possible satisfying the condition?

Answer

To find the coordinates of a point that is equidistant from the two points (-4, 0) and (4, 0), we can follow these steps:

### Step 1: Understand the Concept of Equidistance
We need to find a point (x, y) such that the distance from this point to (-4, 0) is equal to the distance from this point to (4, 0).

### Step 2: Use the Distance Formula
The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

### Step 3: Set Up the Distance Equations
1. Distance from (x, y) to (-4, 0):
\[ d_1 = \sqrt{(x - (-4))^2 + (y - 0)^2} = \sqrt{(x + 4)^2 + y^2} \]

2. Distance from (x, y) to (4, 0):
\[ d_2 = \sqrt{(x - 4)^2 + (y - 0)^2} = \sqrt{(x - 4)^2 + y^2} \]

### Step 4: Set the Distances Equal
Since the point (x, y) is equidistant from both points, we set the distances equal:
\[ \sqrt{(x + 4)^2 + y^2} = \sqrt{(x - 4)^2 + y^2} \]

### Step 5: Square Both Sides
To eliminate the square roots, we square both sides:
\[ (x + 4)^2 + y^2 = (x - 4)^2 + y^2 \]

### Step 6: Simplify the Equation
1. Cancel \(y^2\) from both sides:
\[ (x + 4)^2 = (x - 4)^2 \]

2. Expand both sides:
\[ x^2 + 8x + 16 = x^2 - 8x + 16 \]

3. Cancel \(x^2\) and \(16\) from both sides:
\[ 8x = -8x \]

### Step 7: Solve for x
Combine like terms:
\[ 8x + 8x = 0 \implies 16x = 0 \implies x = 0 \]

### Step 8: Determine the y-coordinate
Since we found \(x = 0\), the y-coordinate can be any value. Therefore, the coordinates of the point can be expressed as:
\[ (0, y) \]
where \(y\) can be any real number.

### Step 9: Conclusion on the Number of Points
Since \(y\) can take any value, there are infinitely many points that are equidistant from the two given points. These points lie on the y-axis.

### Final Answer
The coordinates of the point which is equidistant from the points (-4, 0) and (4, 0) are of the form (0, y), where y is any real number. There are infinitely many such points.

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