In what ratio is the segment joining the points (4, 6) and (–7, −1) divided by x-axis ?

Question

Accepted Answer

To determine the ratio in which the x-axis divides the segment joining the points (4, 6) and (–7, −1), we need to understand the concept of section formula in coordinate geometry. The x-axis divides the line segment at a point where the y-coordinate is zero.

1. **Finding the Point of Intersection:**
   - The line segment joining the points (4, 6) and (–7, −1) will intersect the x-axis where the y-coordinate is zero. 
   - Using the section formula, if a point (x, y) divides the line segment joining (x₁, y₁) and (x₂, y₂) in the ratio m:n, the coordinates of the point are given by:
     $$
     x = \frac{mx₂ + nx₁}{m+n}, \quad y = \frac{my₂ + ny₁}{m+n}
     $$
   - Since the point lies on the x-axis, y = 0. Therefore:
     $$
     \frac{m(-1) + n(6)}{m+n} = 0
     $$
     $$
     -m + 6n = 0
     $$
     $$
     m = 6n
     $$
   - This implies the ratio m:n is 6:1.

2. **Explanation of Option 4:**
   - Option 4 states the ratio is 6:1, which matches our calculation. The x-axis divides the segment in the ratio 6:1, confirming that Option 4 is correct.

3. **Explanation of Incorrect Options:**
   - **Option 1 (1:6):** This suggests the x-axis divides the segment in the ratio 1:6, which contradicts our calculation where m = 6n.
   - **Option 2 (6:2):** This simplifies to 3:1, which is not the correct ratio derived from the section formula.
   - **Option 3 (2:6):** This simplifies to 1:3, which again does not match the calculated ratio of 6:1.

In conclusion, the x-axis divides the segment joining the points (4, 6) and (–7, −1) in the ratio 6:1, making Option 4 the correct answer.