Question

A straight line through the origin \(O\) meets the parallel lines \(4x+2y=9\) and \(2x+y+6=0\) at points \(P\) and \(Q\) respectively. The point \(O\) divides the segment \(PQ\) in the ratio:

• A. \(1:2\)
• B. \(3:4\)
• C. \(2:1\)
• D. \(4:3\)

Hint:

If a variable line through the origin intersects two fixed parallel lines, the ratio in which the origin divides the intercepted segment is \emph{constant} and independent of the slope of the line.

Answer 1

The Correct Option is B

Step 1: Write the equation of the variable line through the origin. Let the line through origin be: \[ y=mx \]
Step 2: Find point \(P\) on the line \(4x+2y=9\). Substitute \(y=mx\): \[ 4x+2mx=9 \] \[ x(4+2m)=9 \Rightarrow x_P=\frac{9}{4+2m} \] Since the line passes through origin, distance \(OP\) is proportional to: \[ OP \propto \left|\frac{9}{4+2m}\right| \]
Step 3: Find point \(Q\) on the line \(2x+y+6=0\). Substitute \(y=mx\): \[ 2x+mx+6=0 \] \[ x(2+m)=-6 \Rightarrow x_Q=\frac{-6}{2+m} \] Distance \(OQ\) is proportional to: \[ OQ \propto \left|\frac{6}{2+m}\right| \]
Step 4: Find the ratio \(OP:OQ\). \[ OP:OQ =\frac{9}{4+2m}:\frac{6}{2+m} \] \[ =9(2+m):6(4+2m) \] Dividing by \(3\): \[ =3(2+m):2(4+2m) \] \[ =\frac{3(2+m)}{4(2+m)}=\frac{3}{4} \]
Hence, the point \(O\) divides the segment \(PQ\) in the ratio \[ \boxed{3:4} \]