A straight line through the origin meets the parallel lines and at points and respectively. Then, the point divides the segment in the ratio
- A.1 : 2
- B.3 : 4
- C.2 : 1
- D.4 : 3
Show answer and solution
Correct option: B
Solution
The two parallel lines are 4x + 2y = 9 (or 2x + y = 9/2) and 2x + y = -6. The perpendicular distances from the origin to these lines are:
- To line 2x + y = 9/2: distance = (9/2)/√5
- To line 2x + y = -6: distance = 6/√5
Since O lies between P and Q (the lines are on opposite sides of the origin — one gives 9/2 > 0 and the other gives -6 < 0), O divides PQ internally. The ratio OP:OQ equals the distance from O to the first line to the distance from O to the second line = (9/2)/√5 : 6/√5 = 9/2 : 6 = 3 : 4.
- To line 2x + y = 9/2: distance = (9/2)/√5
- To line 2x + y = -6: distance = 6/√5
Since O lies between P and Q (the lines are on opposite sides of the origin — one gives 9/2 > 0 and the other gives -6 < 0), O divides PQ internally. The ratio OP:OQ equals the distance from O to the first line to the distance from O to the second line = (9/2)/√5 : 6/√5 = 9/2 : 6 = 3 : 4.
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