A straight line through origin O meets the lines `3y=10-4x` and `8x+6y+5=0` at point A and B respectively. Then , O divides the segment AB in the ratio.

Options

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

Explanation

To solve the problem, we need to find the points of intersection of the line through the origin with the given lines and determine the ratio in which the origin divides the segment AB.

### Step-by-Step Solution:

1. **Identify the Given Lines:**
The equations of the lines are:
- Line 1: \(3y = 10 - 4x\) which can be rewritten as \(4x + 3y - 10 = 0\).
- Line 2: \(8x + 6y + 5 = 0\).

2. **Find the Slope of the Given Lines:**
- For Line 1: Rearranging gives \(y = -\frac{4}{3}x + \frac{10}{3}\). The slope \(m_1 = -\frac{4}{3}\).
- For Line 2: Rearranging gives \(y = -\frac{8}{6}x - \frac{5}{6}\). The slope \(m_2 = -\frac{4}{3}\).

Since both lines have the same slope, they are parallel.

3. **Equation of the Line Through the Origin:**
The line through the origin can be expressed as \(y = mx\) where \(m\) is the slope. We will use \(y = mx\) to find the intersection points A and B.

4. **Find Intersection Point A:**
Substitute \(y = mx\) into the equation of Line 1:
\[ 4x + 3(mx) - 10 = 0 \]
Simplifying gives:
\[ 4x + 3mx - 10 = 0 \implies (4 + 3m)x = 10 \implies x = \frac{10}{4 + 3m} \]
Now substituting \(x\) back to find \(y\):
\[ y = m \left(\frac{10}{4 + 3m}\right) = \frac{10m}{4 + 3m} \]
Thus, point A is:
\[ A\left(\frac{10}{4 + 3m}, \frac{10m}{4 + 3m}\right) \]

5. **Find Intersection Point B:**
Substitute \(y = mx\) into the equation of Line 2:
\[ 8x + 6(mx) + 5 = 0 \]
Simplifying gives:
\[ 8x + 6mx + 5 = 0 \implies (8 + 6m)x = -5 \implies x = \frac{-5}{8 + 6m} \]
Now substituting \(x\) back to find \(y\):
\[ y = m\left(\frac{-5}{8 + 6m}\right) = \frac{-5m}{8 + 6m} \]
Thus, point B is:
\[ B\left(\frac{-5}{8 + 6m}, \frac{-5m}{8 + 6m}\right) \]

6. **Finding the Ratio in Which O Divides AB:**
The coordinates of points A and B are:
\[ A\left(\frac{10}{4 + 3m}, \frac{10m}{4 + 3m}\right), \quad B\left(\frac{-5}{8 + 6m}, \frac{-5m}{8 + 6m}\right) \]
The distances from O to A and O to B can be calculated as:
\[ OA = \sqrt{\left(\frac{10}{4 + 3m}\right)^2 + \left(\frac{10m}{4 + 3m}\right)^2} \]
\[ OB = \sqrt{\left(\frac{-5}{8 + 6m}\right)^2 + \left(\frac{-5m}{8 + 6m}\right)^2} \]

The ratio \(k\) in which O divides AB is given by:
\[ k = \frac{OA}{OB} \]

After substituting and simplifying, we find:
\[ k = \frac{4}{1} \]

### Final Answer:
The origin O divides the segment AB in the ratio \(4:1\).