What is the relation between x and y, if the point P(x, y) is equidistant from the points  A(7,0) and B(0,5)?

Options

A. x + 2y = 9
B. 7x - 5y = 12
C. 5x + 2y = 15
D. 3x - 2y = 7

Answer

To find the relation between \( x \) and \( y \) for the point \( P(x, y) \) that is equidistant from the points \( A(7, 0) \) and \( B(0, 5) \), we will use the distance formula. Here’s the step-by-step solution:

### Step 1: Write the distance formulas
The distance from point \( P(x, y) \) to point \( A(7, 0) \) is given by:
\[ PA = \sqrt{(x - 7)^2 + (y - 0)^2} = \sqrt{(x - 7)^2 + y^2} \]

The distance from point \( P(x, y) \) to point \( B(0, 5) \) is given by:
\[ PB = \sqrt{(x - 0)^2 + (y - 5)^2} = \sqrt{x^2 + (y - 5)^2} \]

### Step 2: Set the distances equal
Since point \( P \) is equidistant from points \( A \) and \( B \), we can set the two distances equal to each other:
\[ \sqrt{(x - 7)^2 + y^2} = \sqrt{x^2 + (y - 5)^2} \]

### Step 3: Square both sides
To eliminate the square roots, we square both sides of the equation:
\[ (x - 7)^2 + y^2 = x^2 + (y - 5)^2 \]

### Step 4: Expand both sides
Now, we expand both sides:
- Left side:
\[ (x - 7)^2 + y^2 = x^2 - 14x + 49 + y^2 \]
- Right side:
\[ x^2 + (y - 5)^2 = x^2 + y^2 - 10y + 25 \]

### Step 5: Set the expanded forms equal
Now we can set the expanded forms equal to each other:
\[ x^2 - 14x + 49 + y^2 = x^2 + y^2 - 10y + 25 \]

### Step 6: Cancel out common terms
We can cancel \( x^2 \) and \( y^2 \) from both sides:
\[ -14x + 49 = -10y + 25 \]

### Step 7: Rearrange the equation
Now, rearranging the equation gives:
\[ -14x + 10y + 49 - 25 = 0 \]
\[ -14x + 10y + 24 = 0 \]

### Step 8: Simplify the equation
To simplify, we can divide the entire equation by 2:
\[ -7x + 5y + 12 = 0 \]

### Step 9: Rearranging to standard form
Rearranging gives us:
\[ 7x - 5y = 12 \]

Thus, the relation between \( x \) and \( y \) is:
\[ 7x - 5y = 12 \]