Find the ratio in which the line segment joining the points (−3, 10) and (6, −8) is divided by (−1, 6).
2 : 7
Let the point \((-1, 6)\) divide the line segment joining \((-3, 10)\) and \((6, -8)\) in the ratio \(m : n\).
Using the section formula for internal division, the coordinates of the dividing point are
\( \left( \frac{m x_2 + n x_1}{m + n}, \, \frac{m y_2 + n y_1}{m + n} \right). \)
Here, \(x_1 = -3\), \(y_1 = 10\), \(x_2 = 6\), \(y_2 = -8\), and the dividing point is \((-1, 6)\).
So for the x-coordinate:
\( -1 = \frac{m(6) + n(-3)}{m + n} \).
Cross-multiplying gives: \( -1(m + n) = 6m - 3n \) → \( -m - n = 6m - 3n \).
Rearranging: \( -m - n - 6m + 3n = 0 \) → \( -7m + 2n = 0 \).
Thus, \( 2n = 7m \) → \( \frac{m}{n} = \frac{2}{7} \).
This ratio also satisfies the y-coordinate equation, so the division ratio is \(2 : 7\).