Question

If \( (a, -2a), \, a>0 \) is the midpoint of a line segment intercepted between the co-ordinate axes, then the equation of the line is

• A. \( x - 2y + 4a = 0 \)
• B. \( 2x - y = 4a \)
• C. \( x - 2y = 5a \)
• D. \( 2x - y + 4a = 0 \)

Hint:

To find the equation of a line passing through the intercepts, use the midpoint formula to find the intercepts and then apply the slope formula.

Answer 1

The Correct Option is B

Step 1: Understanding the midpoint condition.
We are given that the point \( (a, -2a) \) is the midpoint of a line segment intercepted between the coordinate axes. Let the intercepts on the x-axis and y-axis be \( A(x_1, 0) \) and \( B(0, y_1) \), respectively. Since the midpoint is \( (a, -2a) \), we use the midpoint formula to find the equation of the line.

Step 2: Using the midpoint formula.
The midpoint of \( A(x_1, 0) \) and \( B(0, y_1) \) is: \[ \left( \frac{x_1 + 0}{2}, \frac{0 + y_1}{2} \right) = (a, -2a) \] From this, we find: \[ \frac{x_1}{2} = a \quad \Rightarrow \quad x_1 = 2a, \quad \frac{y_1}{2} = -2a \quad \Rightarrow \quad y_1 = -4a \]
Step 3: Finding the equation of the line.
The equation of the line passing through \( A(2a, 0) \) and \( B(0, -4a) \) can be written in slope-intercept form. The slope is: \[ \text{slope} = \frac{-4a - 0}{0 - 2a} = 2 \] Thus, the equation of the line is: \[ y - 0 = 2(x - 2a) \] Simplifying this equation, we get: \[ 2x - y = 4a \]
Step 4: Conclusion.
Thus, the equation of the line is \( 2x - y = 4a \), which makes option (B) the correct answer.