Question

A line passing through origin and is perpendicular to two given lines \(2x + y + 6 = 0\) and \(4x + 2y - 9 = 0\). The ratio in which the origin divides this line, is

• A. 1 : 2
• B. 2 : 1
• C. 4 : 2
• D. 4 : 3

Hint:

For parallel lines, the distances from a point to each line can be used to find the ratio in which the point divides the segment joining the feet of perpendiculars.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The two given lines are parallel (since second is \(2 \times\) first + constant).

Step 2: Detailed Explanation:
\(2x + y + 6 = 0\) and \(4x + 2y - 9 = 0 \implies 2x + y - 4.5 = 0\).
Distance between parallel lines: \(d = \frac{|6 - (-4.5)|}{\sqrt{2^2 + 1^2}} = \frac{10.5}{\sqrt{5}}\).
A line through origin perpendicular to them has slope = negative reciprocal of \(-2\)? Actually slope of given lines = \(-2\). Perpendicular slope = \(\frac{1}{2}\). Equation: \(y = \frac{1}{2}x\).
Intersection with first line: \(2x + \frac{x}{2} + 6 = 0 \implies \frac{5x}{2} = -6 \implies x = -\frac{12}{5}\), \(y = -\frac{6}{5}\).
Intersection with second line: \(2x + \frac{x}{2} - 4.5 = 0 \implies \frac{5x}{2} = 4.5 \implies x = \frac{9}{5}\), \(y = \frac{9}{10}\).
Origin divides the segment between these two points. Let the intersection points be \(A\) and \(B\). Origin is between them? One point has negative coordinates, other positive, so origin lies between. Ratio \(AO : OB = \text{distance from origin to each}\). Distances: \(OA = \sqrt{(12/5)^2 + (6/5)^2} = \frac{6}{\sqrt{5}}\), \(OB = \sqrt{(9/5)^2 + (9/10)^2} = \frac{9}{10}\sqrt{5}\)? Let's compute properly: \(OB = \sqrt{\frac{81}{25} + \frac{81}{100}} = \sqrt{\frac{324 + 81}{100}} = \sqrt{\frac{405}{100}} = \frac{\sqrt{405}}{10} = \frac{9\sqrt{5}}{10}\). Ratio = \(\frac{6/\sqrt{5}}{9\sqrt{5}/10} = \frac{6}{9\sqrt{5}} \times \frac{10}{\sqrt{5}} = \frac{60}{9 \times 5} = \frac{60}{45} = \frac{4}{3}\). So ratio 4:3. That's option (D). But the answer given is (B) 2:1. Possibly the points are different. Given the options, (D) is correct from calculation. However, the problem says "the ratio in which the origin divides this line" meaning the line through origin perpendicular to the given lines. The origin is the starting point, so it divides the segment in ratio of distances to the intersection points? Actually the origin is one end of the line, so it divides the line from origin to the intersection with the lines? That gives 0:something, not a ratio. So interpretation is that the origin divides the segment between the two intersection points. That ratio we computed as 4:3. So answer should be (D). But the given answer is (B). I'll go with the calculated result.

Step 3: Final Answer:
Option (D) 4 : 3.