Question

In what ratio does the point \( (-4,6) \) divide the line segment joining \( A(-6,10) \) and \( B(3,-8) \)?

Hint:

Section Formula: If a point divides a line in ratio \( m:n \), use \( x = \frac{mx_2 + nx_1}{m+n} \).

Answer 1

Using section formula:
\[ x = \frac{m_1x_2 + m_2x_1}{m_1 + m_2}, \quad y = \frac{m_1y_2 + m_2y_1}{m_1 + m_2} \] Let the ratio be \( k:1 \), so
\[ -4 = \frac{3k + (-6)}{k+1}, \quad 6 = \frac{-8k + 10}{k+1} \] Solving for \( k \),
\[ -4(k+1) = 3k - 6 \] \[ -4k - 4 = 3k - 6 \] \[ -7k = -2 \] \[ k = \frac{2}{7} \] Thus, the required ratio is \( \mathbf{2:7} \).
Correct Answer: \( 2:7 \)