The equation of a line bisecting the join of (2010, 1600) and `(-1340, 1080)` and having intercept on the axes in the ratio 1 : 2 is

Options

A. `2x+y=1680`
B. `x+2y=1680`
C. `2x+y=2010`
D. none of these

Answer

To find the equation of the line bisecting the segment joining the points (2010, 1600) and (-1340, 1080) with intercepts on the axes in the ratio 1:2, we can follow these steps:

### Step 1: Find the Midpoint of the Line Segment
The midpoint \( M \) of the line segment joining points \( A(2010, 1600) \) and \( B(-1340, 1080) \) can be calculated using the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates:
\[ M = \left( \frac{2010 + (-1340)}{2}, \frac{1600 + 1080}{2} \right) \]
Calculating the coordinates:
\[ M = \left( \frac{2010 - 1340}{2}, \frac{1600 + 1080}{2} \right) = \left( \frac{670}{2}, \frac{2680}{2} \right) = (335, 1340) \]

### Step 2: Determine the Intercepts
Let the intercepts on the axes be \( k \) and \( 2k \). The points on the axes would be \( (k, 0) \) and \( (0, 2k) \).

### Step 3: Use the Intercept Form of the Line
The equation of the line in intercept form is given by:
\[ \frac{x}{k} + \frac{y}{2k} = 1 \]
Multiplying through by \( 2k \) gives:
\[ 2y + 2x = 2k \]
or
\[ 2x + y = 2k \]

### Step 4: Substitute the Midpoint into the Line Equation
Since the line passes through the midpoint \( (335, 1340) \), we substitute these values into the equation:
\[ 2(335) + 1340 = 2k \]
Calculating:
\[ 670 + 1340 = 2k \implies 2010 = 2k \implies k = 1005 \]

### Step 5: Write the Final Equation
Now substituting \( k = 1005 \) back into the line equation:
\[ 2x + y = 2(1005) \implies 2x + y = 2010 \]

Thus, the equation of the line is:
\[ 2x + y = 2010 \]

### Final Answer
The equation of the line is:
\[ \boxed{2x + y = 2010} \]