Step 1: Understanding the Concept:
We have two lines and a third line passing through the origin intersecting them.
First, let's analyze the given two lines.
Line 1: \( 4x + 3y - 10 = 0 \)
Line 2: \( 8x + 6y + 5 = 0 \implies 4x + 3y + \frac{5}{2} = 0 \)
Notice that the coefficients of \( x \) and \( y \) are proportional. Thus, these two lines are parallel.
Step 2: Key Formula or Approach:
When a transversal line (in this case, the line through the origin) intersects two parallel lines, the ratio of the distances from any point on the transversal to the intersection points is proportional to the perpendicular distances from that point to the parallel lines.
Distance of origin from a line \( ax + by + c = 0 \) is \( d = \frac{|c|}{\sqrt{a^2 + b^2}} \).
Step 3: Detailed Explanation:
Let the perpendicular distance from the origin \( O(0,0) \) to Line 1 be \( d_1 \):
\[ d_1 = \frac{|4(0) + 3(0) - 10|}{\sqrt{4^2 + 3^2}} = \frac{|-10|}{5} = 2 \]
Let the perpendicular distance from the origin \( O(0,0) \) to Line 2 be \( d_2 \):
\[ d_2 = \frac{|8(0) + 6(0) + 5|}{\sqrt{8^2 + 6^2}} = \frac{5}{\sqrt{100}} = \frac{5}{10} = \frac{1}{2} \]
Let the line through the origin be transversal \( t \), meeting Line 1 at \( A \) and Line 2 at \( B \).
Since Line 1 and Line 2 are parallel, the triangles formed by the origin, the intersection points \( A, B \), and the feet of the perpendiculars are similar.
Therefore, the ratio of the segment lengths \( OA \) and \( OB \) is equal to the ratio of their respective perpendicular distances from the origin.
\[ \frac{OA}{OB} = \frac{d_1}{d_2} \]
\[ \frac{OA}{OB} = \frac{2}{1/2} = 4 \]
So, the origin \( O \) divides the segment \( AB \) in the ratio \( OA : OB = 4 : 1 \).
Since the constant terms (-10 and +5) have opposite signs when the equations are written as \( ax+by=c \), the origin lies between the two lines, making it an internal division.
Step 4: Final Answer:
The ratio is \( 4 : 1 \).