A line passes through P(1,2) such that the portion of the line intercepted between the axes is bisected at P. The equation of the line is

Options

A. `x+2y-5=0`
B. `x-y+1=0`
C. `x+y-3=0`
D. `2x+y-4=0`

Answer

To find the equation of the line that passes through the point P(1, 2) and is bisected at P by the portion of the line intercepted between the axes, we can follow these steps:

### Step 1: Understand the Intercept Form of the Line
The intercept form of a line is given by the equation:
\[ \frac{x}{a} + \frac{y}{b} = 1 \]
where \(a\) is the x-intercept and \(b\) is the y-intercept of the line.

### Step 2: Determine the Intercepts
Since the line is bisected at point P(1, 2), we know that P is the midpoint of the intercepts on the axes. Let the x-intercept be \(a\) and the y-intercept be \(b\).

### Step 3: Use the Midpoint Formula
The midpoint of the intercepts (which are at (a, 0) and (0, b)) is given by:
\[ \left(\frac{a + 0}{2}, \frac{0 + b}{2}\right) = \left(\frac{a}{2}, \frac{b}{2}\right) \]
Since this midpoint is equal to P(1, 2), we can set up the following equations:
\[ \frac{a}{2} = 1 \quad \text{and} \quad \frac{b}{2} = 2 \]

### Step 4: Solve for a and b
From the first equation:
\[ a = 2 \times 1 = 2 \]
From the second equation:
\[ b = 2 \times 2 = 4 \]

### Step 5: Write the Equation of the Line
Now that we have the values of \(a\) and \(b\), we can substitute them back into the intercept form of the line:
\[ \frac{x}{2} + \frac{y}{4} = 1 \]

### Step 6: Simplify the Equation
To eliminate the fractions, we can multiply through by 4 (the least common multiple of the denominators):
\[ 4 \left(\frac{x}{2}\right) + 4 \left(\frac{y}{4}\right) = 4 \]
This simplifies to:
\[ 2x + y = 4 \]

### Step 7: Rearrange to Standard Form
Rearranging the equation gives us:
\[ 2x + y - 4 = 0 \]

### Final Answer
Thus, the equation of the line is:
\[ \boxed{2x + y - 4 = 0} \]