Question

The equation of a line passing through the point of intersection of the lines \[ x + 2y + 8 = 0 \quad \text{and} \quad 3x - y + 4 = 0 \] and having x– and y–intercepts zero is

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

Hint:

A line with zero x– and y–intercepts always passes through the origin and has equation \(ax + by = 0\).

Answer 1

The Correct Option is B

Step 1: Find the point of intersection.
Solve the given equations:
\[ x + 2y + 8 = 0 \quad (1) \] \[ 3x - y + 4 = 0 \quad (2) \] From (2), \[ y = 3x + 4 \] Substitute in (1):
\[ x + 2(3x + 4) + 8 = 0 \Rightarrow 7x + 16 = 0 \Rightarrow x = -\frac{16}{7} \] \[ y = 3\left(-\frac{16}{7}\right) + 4 = -\frac{20}{7} \] Step 2: Use condition of zero intercepts.
A line with zero x– and y–intercepts passes through the origin, hence its equation is of the form \[ ax + by = 0 \] Step 3: Use point condition.
Substitute the intersection point \(\left(-\frac{16}{7}, -\frac{20}{7}\right)\):
\[ a(-16) + b(-20) = 0 \Rightarrow 16a = 20b \Rightarrow 4a = 5b \] Step 4: Write the required equation.
Taking \(a = 5\) and \(b = 4\), we get \[ 5x - 4y = 0 \]