Question

In what ratio does the point \( (-1,6) \) divide the line segment joining the points \( (-3,10) \) and \( (6,-8) \)?

Hint:

Use the section formula to divide a line segment in a given ratio.

Answer 1

Using the section formula:
\[ (x, y) = \left( \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}, \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2} \right). \] Let the ratio be \( m:n \).
\[ (-1,6) = \left( \frac{6m + (-3n)}{m+n}, \frac{-8m + 10n}{m+n} \right). \] Equating \( x \)-coordinates:
\[ \frac{6m - 3n}{m+n} = -1. \] \[ 6m - 3n = -m - n. \] \[ 6m + m = 3n - n. \] \[ 7m = 2n. \] \[ \frac{m}{n} = \frac{2}{7}. \] Thus, the required ratio is \( 2:7 \).