The point on x-axis whichis equidistant from (-4, 0) and (10, 0) is:

Options

A. (7, 0)
B. (5, 0)
C. (0, 0)
D. (3, 0)

Answer

To find the point on the x-axis that is equidistant from the points (-4, 0) and (10, 0), we can follow these steps:

### Step 1: Define the point on the x-axis
Let the point on the x-axis be \( P(x, 0) \).

### Step 2: Use the distance formula
The distance from point \( P \) to point \( A(-4, 0) \) is given by:
\[ d(P, A) = \sqrt{(x - (-4))^2 + (0 - 0)^2} = \sqrt{(x + 4)^2} \]
This simplifies to:
\[ d(P, A) = |x + 4| \]

The distance from point \( P \) to point \( B(10, 0) \) is given by:
\[ d(P, B) = \sqrt{(x - 10)^2 + (0 - 0)^2} = \sqrt{(x - 10)^2} \]
This simplifies to:
\[ d(P, B) = |x - 10| \]

### Step 3: Set the distances equal
Since point \( P \) is equidistant from points \( A \) and \( B \), we can set the distances equal to each other:
\[ |x + 4| = |x - 10| \]

### Step 4: Solve the absolute value equation
We will consider two cases based on the definition of absolute values.

**Case 1:** \( x + 4 = x - 10 \)
\[ 4 = -10 \quad \text{(This case is not possible)} \]

**Case 2:** \( x + 4 = -(x - 10) \)
\[ x + 4 = -x + 10 \]
Adding \( x \) to both sides:
\[ 2x + 4 = 10 \]
Subtracting 4 from both sides:
\[ 2x = 6 \]
Dividing by 2:
\[ x = 3 \]

### Step 5: Write the coordinates of point P
Since \( P \) is on the x-axis, its coordinates are:
\[ P(3, 0) \]

### Conclusion
The point on the x-axis which is equidistant from (-4, 0) and (10, 0) is \( (3, 0) \).

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