If a point (x,y) is equidistant from the points A(9,8) and B(17,8), then the relation between x and y is :

Options

A. `x+y=13`
B. `x-13=0`
C. `y-13=0`
D. `x-y=13`

Answer

To find the relation between \( x \) and \( y \) for the point \( (x, y) \) that is equidistant from points \( A(9, 8) \) and \( B(17, 8) \), we can follow these steps:

### Step 1: Write the distance formulas
The distance from point \( (x, y) \) to point \( A(9, 8) \) is given by:
\[ d_A = \sqrt{(x - 9)^2 + (y - 8)^2} \]
The distance from point \( (x, y) \) to point \( B(17, 8) \) is given by:
\[ d_B = \sqrt{(x - 17)^2 + (y - 8)^2} \]

### Step 2: Set the distances equal
Since the point \( (x, y) \) is equidistant from points \( A \) and \( B \), we set the distances equal to each other:
\[ \sqrt{(x - 9)^2 + (y - 8)^2} = \sqrt{(x - 17)^2 + (y - 8)^2} \]

### Step 3: Square both sides
To eliminate the square roots, we square both sides of the equation:
\[ (x - 9)^2 + (y - 8)^2 = (x - 17)^2 + (y - 8)^2 \]

### Step 4: Simplify the equation
Notice that \( (y - 8)^2 \) cancels out from both sides:
\[ (x - 9)^2 = (x - 17)^2 \]

### Step 5: Expand both sides
Now, expand both sides:
\[ (x^2 - 18x + 81) = (x^2 - 34x + 289) \]

### Step 6: Cancel \( x^2 \) and rearrange
Subtract \( x^2 \) from both sides:
\[ -18x + 81 = -34x + 289 \]
Now, rearranging gives:
\[ 34x - 18x = 289 - 81 \]
\[ 16x = 208 \]

### Step 7: Solve for \( x \)
Now, divide both sides by 16:
\[ x = \frac{208}{16} = 13 \]

### Step 8: Establish the relation
Since \( y \) does not appear in the equation derived from the distances, we conclude that \( y \) can take any value. Therefore, the relation between \( x \) and \( y \) is:
\[ x = 13 \]

### Final Answer
The relation between \( x \) and \( y \) is \( x - 13 = 0 \).

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