The equation to the line bisecting the joining of (3. - 4) and (5, 2) having its intercepts on X-axis and Y-axis in the ratio 2:1, is

Options

A. x + y - 3= 0
B. 2x - y = 9
C. x + 2y = 2
D. 2x + y = 7

Answer

To find the equation of the line that bisects the line segment joining the points (3, -4) and (5, 2) and has intercepts on the X-axis and Y-axis in the ratio 2:1, we can follow these steps:

### Step 1: Find the Midpoint of the Line Segment
The midpoint \( M \) of the line segment joining the points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates \( (3, -4) \) and \( (5, 2) \):
\[ M = \left( \frac{3 + 5}{2}, \frac{-4 + 2}{2} \right) = \left( \frac{8}{2}, \frac{-2}{2} \right) = (4, -1) \]

### Step 2: Determine the Intercept Form of the Line
The intercept form of the equation of a line is given by:
\[ \frac{x}{a} + \frac{y}{b} = 1 \]
where \( a \) is the x-intercept and \( b \) is the y-intercept. Given that the intercepts are in the ratio \( 2:1 \), we can express them as:
\[ a = 2k \quad \text{and} \quad b = k \]

### Step 3: Substitute the Intercepts into the Equation
Substituting \( a \) and \( b \) into the intercept form:
\[ \frac{x}{2k} + \frac{y}{k} = 1 \]

### Step 4: Substitute the Midpoint into the Equation
Since the line passes through the midpoint \( (4, -1) \), we substitute these coordinates into the line equation:
\[ \frac{4}{2k} + \frac{-1}{k} = 1 \]

### Step 5: Solve for \( k \)
To solve for \( k \), we first find a common denominator:
\[ \frac{4 - 2}{2k} = 1 \implies \frac{2}{2k} = 1 \implies \frac{1}{k} = 1 \implies k = 1 \]

### Step 6: Find the Values of \( a \) and \( b \)
Now substituting \( k = 1 \) back into the expressions for \( a \) and \( b \):
\[ a = 2k = 2 \cdot 1 = 2 \quad \text{and} \quad b = k = 1 \]

### Step 7: Write the Final Equation
Substituting \( a \) and \( b \) into the intercept form:
\[ \frac{x}{2} + \frac{y}{1} = 1 \]
Multiplying through by 2 to eliminate the fractions:
\[ x + 2y = 2 \]

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