Find the ratio in which the line segment joining the points (-3, 10) and (6, -8) is divided by (-1, 6).
2:7
Explanation
This problem mirrors Example 7 from the textbook, where the section formula is used to find an unknown ratio. By letting the ratio be k:1 and equating the coordinates obtained from the formula to the given point, we solve for k. The positive value confirms internal division, and the resulting ratio 2:7 indicates that the point is closer to A(-3,10) than to B(6,-8).
Solution Steps
Let the point (-1, 6) divide the segment joining A(-3, 10) and B(6, -8) in the ratio k : 1.
Using the section formula: x-coordinate = (k·6 + 1·(-3))/(k+1) = (6k - 3)/(k+1), y-coordinate = (k·(-8) + 1·10)/(k+1) = (-8k + 10)/(k+1).
Set these equal to the given coordinates: (6k - 3)/(k+1) = -1 and (-8k + 10)/(k+1) = 6.
Solve the first equation: 6k - 3 = -k - 1 ⇒ 7k = 2 ⇒ k = 2/7.
Verify with the second equation: (-8·(2/7) + 10)/(2/7 + 1) = ( -16/7 + 70/7 )/(9/7) = (54/7)/(9/7) = 6, which matches.
Thus the ratio is k : 1 = 2/7 : 1 = 2 : 7.