Question

A straight line through the origin \( O \) meets the line \( 3y = 10 - 4x \) and \( 8x + 6y + 5 = 0 \) at the points \( A \) and \( B \) respectively. Then \( O \) divides the segment \( AB \) in the ratio

• A. \( 4 : 1 \)
• B. \( 2 : 3 \)
• C. \( 1 : 5 \)
• D. \( 1 : 3 \)

Hint:

For parallel lines, the intercept ratio on any transversal through the origin is the ratio of their perpendicular distances.

Answer 1

The Correct Option is A

Step 1: Concept The distance of the origin from lines and the concept of similar triangles or ratios of segments along a line.

Step 2: Meaning The given lines are \(4x + 3y - 10 = 0\) and \(8x + 6y + 5 = 0\). Notice they are parallel (\(4x+3y = 10\) and \(2(4x+3y) = -5\)).

Step 3: Analysis
Distance of origin from \(L_1\): \(d_1 = \frac{|-10|}{\sqrt{4^2+3^2}} = 2\).
Distance of origin from \(L_2\): \(d_2 = \frac{|5|}{\sqrt{8^2+6^2}} = \frac{5}{10} = 0.5\).
Since the line passes through the origin, the ratio \(OA:OB\) is the ratio of their perpendicular distances.


Step 4: Conclusion Ratio = \(\frac{2}{0.5} = \frac{4}{1}\). Since origin is between them (one constant is negative, one positive), it's a \(4:1\) division. Final Answer: (A)