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. Then the point O divides the segment PQ in the ratio

• A. 1:2
• B. 3:4
• C. 2:1
• D. 4:3

Answer 1

The correct option is B 3:4

The given lines are
$2x+y=9/2\left(1\right)$
and $2x+y=-6\left(2\right)$
Signs of constants on R.H.S. show that two lines lie on opp. sides of origin. Let any line through origin meets these lines in P and Q respectively, then the required ratio is OP: OQ

Now in$\Delta OPAand\Delta OQC,$
$\angle POA=\angle QOC(ver.opp.{\angle }^{\prime }s)$
$\angle PAO=\angle OCQ(alt.int.{\angle }^{\prime }s)$
$\therefore \Delta OPA$ ~$\Delta OQC$ (by AAA similarity)
$\therefore \frac{OP}{OQ}=\frac{OA}{OC}=\frac{9/4}{3}=\frac{3}{4}$
$\therefore$ Req.ratio is $3:4.$