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Step-by-Step Solution
Step 1: Rewrite the Given Lines in Standard Form
The first given line is 3y = 10 − 4x. Rewriting in standard linear form, we get:
$4x + 3y - 10 = 0$.
The second given line is already in standard form:
$8x + 6y + 5 = 0$.
Step 2: Observe that the Lines are Parallel
To check for parallelism, compare the coefficients of x and y in both equations. The first line has coefficients (4, 3), and the second line has coefficients (8, 6). We note:
$ \frac{4}{8} = \frac{3}{6} = \frac{1}{2}.
Since these ratios are equal, the lines are indeed parallel.
Step 3: Use Perpendicular Distances to Find the Ratio
Let O be the origin (0, 0). We want to find the ratio in which O divides AB, where A is on $4x + 3y - 10 = 0$ and B is on $8x + 6y + 5 = 0$.
When a line through O meets two parallel lines, the ratio in which O divides the segment between the points of intersection is equal to the ratio of the perpendicular distances from O to these lines.
Step 4: Compute the Perpendicular Distance from O to the First Line
The perpendicular distance $P_1$ from (0, 0) to $4x + 3y - 10 = 0$ is given by:
$
P_1 = \left| \frac{4(0) + 3(0) - 10}{\sqrt{4^2 + 3^2}} \right|
= \left| \frac{-10}{\sqrt{16 + 9}} \right|
= \frac{10}{5}
= 2.
$
Step 5: Compute the Perpendicular Distance from O to the Second Line
The perpendicular distance $P_2$ from (0, 0) to $8x + 6y + 5 = 0$ is given by:
$
P_2 = \left| \frac{8(0) + 6(0) + 5}{\sqrt{8^2 + 6^2}} \right|
= \left| \frac{5}{\sqrt{64 + 36}} \right|
= \frac{5}{10}
= \frac{1}{2}.
$
Step 6: Determine the Required Ratio
The ratio in which the origin O divides AB is the same as the ratio of these perpendicular distances $P_1 : P_2$.
$
P_1 : P_2 = 2 : \frac{1}{2} = 2 \div \frac{1}{2} = 4 : 1.
$
Therefore, O divides the segment AB in the ratio 4 : 1.
