Question

Find the ratio in which the point \( (-1, 6) \) divides the line segment joining the points \( (-3, 10) \) and \( (6, -8) \).

Hint:

To find the ratio in which a point divides a line segment, use the section formula and solve the system of equations for the x and y coordinates.

Answer 1

Let the required ratio be \( m:n \). The section formula gives the coordinates of the point dividing the line segment joining two points \( (x_1, y_1) \) and \( (x_2, y_2) \) in the ratio \( m:n \) as: \[ x = \frac{m x_2 + n x_1}{m + n}, \quad y = \frac{m y_2 + n y_1}{m + n}. \] Given: - Point \( A(-3, 10) \), - Point \( B(6, -8) \), - The point dividing the segment is \( P(-1, 6) \). Let the ratio be \( m:n \). The coordinates of \( P \) are: \[ x = \frac{m \times 6 + n \times (-3)}{m + n}, \quad y = \frac{m \times (-8) + n \times 10}{m + n}. \] For the x-coordinate: \[ -1 = \frac{6m - 3n}{m + n}. \] For the y-coordinate: \[ 6 = \frac{-8m + 10n}{m + n}. \] Now, solve the two equations. Start with the x-coordinate equation: \[ -1(m + n) = 6m - 3n \quad \Rightarrow \quad -m - n = 6m - 3n \quad \Rightarrow \quad -m - n - 6m + 3n = 0 \quad \Rightarrow \quad -7m + 2n = 0 \quad \Rightarrow \quad 7m = 2n. \] Thus, the ratio is: \[ \frac{m}{n} = \frac{2}{7}. \]
Conclusion: The ratio in which the point \( (-1, 6) \) divides the line segment is \( 2:7 \).