A straight line passes through the point (3, 2) and the portion of this line, intercepted between the positive axes, is bisected at this point. Find the equation of the line. 

Answer

To find the equation of the line that passes through the point (3, 2) and is bisected at this point, we can follow these steps:

### Step 1: Understand the Problem
We need to find the equation of a line that intercepts the positive x-axis and y-axis and has the point (3, 2) as the midpoint of these intercepts.

### Step 2: Define the Intercepts
Let the x-intercept be \( A(a, 0) \) and the y-intercept be \( B(0, b) \). Since the point (3, 2) is the midpoint of the segment \( AB \), we can use the midpoint formula.

### Step 3: Use the Midpoint Formula
The midpoint \( P \) of points \( A(a, 0) \) and \( B(0, b) \) is given by:
\[ P = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{a + 0}{2}, \frac{0 + b}{2} \right) = \left( \frac{a}{2}, \frac{b}{2} \right) \]
Since \( P = (3, 2) \), we can set up the following equations:
\[ \frac{a}{2} = 3 \quad \text{and} \quad \frac{b}{2} = 2 \]

### Step 4: Solve for \( a \) and \( b \)
From the first equation:
\[ a = 3 \times 2 = 6 \]
From the second equation:
\[ b = 2 \times 2 = 4 \]

### Step 5: Identify the Intercepts
Now we have the intercepts:
- \( A(6, 0) \) (x-intercept)
- \( B(0, 4) \) (y-intercept)

### Step 6: Use the Two-Point Form of the Line
The two-point form of the equation of a line passing through points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1) \]
Substituting \( A(6, 0) \) and \( B(0, 4) \):
\[ y - 0 = \frac{4 - 0}{0 - 6} (x - 6) \]
This simplifies to:
\[ y = \frac{4}{-6} (x - 6) \]
\[ y = -\frac{2}{3} (x - 6) \]

### Step 7: Rearranging the Equation
Distributing the right side:
\[ y = -\frac{2}{3}x + 4 \]
To write it in standard form, we can multiply through by 3 to eliminate the fraction:
\[ 3y = -2x + 12 \]
Rearranging gives:
\[ 2x + 3y - 12 = 0 \]

### Final Equation
Thus, the equation of the line is:
\[ 2x + 3y = 12 \]

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