What is the value of a if the points (3, 5) and (7, 1) are equidistant from the point (a, 0) ?

Answer

To find the value of \( a \) such that the points \( (3, 5) \) and \( (7, 1) \) are equidistant from the point \( (a, 0) \), we can follow these steps:

### Step 1: Set up the distance equations
We will use the distance formula, which states that the distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Let \( d_1 \) be the distance from \( (a, 0) \) to \( (3, 5) \) and \( d_2 \) be the distance from \( (a, 0) \) to \( (7, 1) \).

### Step 2: Calculate \( d_1 \)
Using the distance formula for \( d_1 \):
\[ d_1 = \sqrt{(3 - a)^2 + (5 - 0)^2} = \sqrt{(3 - a)^2 + 25} \]

### Step 3: Calculate \( d_2 \)
Using the distance formula for \( d_2 \):
\[ d_2 = \sqrt{(7 - a)^2 + (1 - 0)^2} = \sqrt{(7 - a)^2 + 1} \]

### Step 4: Set the distances equal
Since the points are equidistant from \( (a, 0) \), we set \( d_1 = d_2 \):
\[ \sqrt{(3 - a)^2 + 25} = \sqrt{(7 - a)^2 + 1} \]

### Step 5: Square both sides to eliminate the square roots
Squaring both sides gives:
\[ (3 - a)^2 + 25 = (7 - a)^2 + 1 \]

### Step 6: Expand both sides
Expanding both sides:
\[ (3 - a)^2 = 9 - 6a + a^2 \]
\[ (7 - a)^2 = 49 - 14a + a^2 \]
So we have:
\[ 9 - 6a + a^2 + 25 = 49 - 14a + a^2 + 1 \]

### Step 7: Simplify the equation
Combining like terms:
\[ 34 - 6a = 50 - 14a \]

### Step 8: Rearrange the equation
Rearranging gives:
\[ 14a - 6a = 50 - 34 \]
\[ 8a = 16 \]

### Step 9: Solve for \( a \)
Dividing both sides by 8:
\[ a = \frac{16}{8} = 2 \]

### Final Answer
The value of \( a \) is \( 2 \).

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