A straight line passes through the point P(3, 2). It meets the x-axis at point A and the y-axis at point B. If `(PA)/(PB) = 2/3 `. find the equation of the line that passes through the point P and is perpendicular to line AB.

Answer

To find the equation of the line that passes through the point P(3, 2) and is perpendicular to line AB, we will follow these steps:

### Step 1: Identify Points A and B
Let the coordinates of point A (where the line meets the x-axis) be (a, 0) and the coordinates of point B (where the line meets the y-axis) be (0, b).

### Step 2: Use the Ratio PA/PB
According to the problem, the ratio of the distances PA to PB is given as \( \frac{PA}{PB} = \frac{2}{3} \). This means:
- PA = 2k
- PB = 3k
for some positive constant k.

### Step 3: Calculate the Lengths PA and PB
Using the distance formula, we can express PA and PB in terms of a and b:
- \( PA = \sqrt{(3 - a)^2 + (2 - 0)^2} = \sqrt{(3 - a)^2 + 4} \)
- \( PB = \sqrt{(3 - 0)^2 + (2 - b)^2} = \sqrt{9 + (2 - b)^2} \)

### Step 4: Set Up the Equation from the Ratio
From the ratio given, we have:
\[ \frac{\sqrt{(3 - a)^2 + 4}}{\sqrt{9 + (2 - b)^2}} = \frac{2}{3} \]
Squaring both sides gives:
\[ \frac{(3 - a)^2 + 4}{9 + (2 - b)^2} = \frac{4}{9} \]

### Step 5: Cross Multiply and Simplify
Cross-multiplying gives:
\[ 9((3 - a)^2 + 4) = 4(9 + (2 - b)^2) \]
Expanding both sides:
\[ 9(3 - a)^2 + 36 = 36 + 4(2 - b)^2 \]
This simplifies to:
\[ 9(3 - a)^2 = 4(2 - b)^2 \]

### Step 6: Express a and b in terms of k
From the ratio of the segments, we can express:
- \( a = 3 + \frac{2}{5}(3) = 5 \)
- \( b = 2 + \frac{3}{5}(2) = 5 \)

### Step 7: Find the Slope of Line AB
Now we can find the slope of line AB using points A(5, 0) and B(0, 5):
\[ m_1 = \frac{0 - 5}{5 - 0} = -1 \]

### Step 8: Find the Slope of the Perpendicular Line
Since the line we are looking for is perpendicular to line AB, its slope \( m_2 \) will be the negative reciprocal of \( m_1 \):
\[ m_2 = 1 \]

### Step 9: Write the Equation of the Line
Using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Substituting \( P(3, 2) \) and \( m_2 = 1 \):
\[ y - 2 = 1(x - 3) \]
This simplifies to:
\[ y - 2 = x - 3 \implies x - y - 1 = 0 \]

### Final Answer
The equation of the line that passes through point P and is perpendicular to line AB is:
\[ x - y - 1 = 0 \]