Question

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

Hint:

Always apply the section formula carefully for both \(x\) and \(y\) coordinates to verify your ratio.

Answer 1

Step 1: Recall the section formula.
If a point \(P(x, y)\) divides the line joining \(A(x_1, y_1)\) and \(B(x_2, y_2)\) in the ratio \(m:n\), then \[ (x, y) = \left( \frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n} \right) \]
Step 2: Substitute the coordinates.
\[ x = -4, \; y = 6, \; A(-6, 10), \; B(3, -8) \] Substitute in the formula: \[ -4 = \frac{m(3) + n(-6)}{m + n}, \quad 6 = \frac{m(-8) + n(10)}{m + n} \]
Step 3: Simplify the first equation.
\[ -4(m + n) = 3m - 6n \Rightarrow -4m - 4n = 3m - 6n \] \[ -7m = -2n \Rightarrow \frac{m}{n} = \frac{2}{7} \]
Step 4: Verification using the second equation.
\[ 6(m + n) = -8m + 10n \Rightarrow 6m + 6n = -8m + 10n \] \[ 14m = 4n \Rightarrow \frac{m}{n} = \frac{2}{7} \]
Step 5: Conclusion.
Hence, the point \((-4, 6)\) divides the line segment joining \(A\) and \(B\) in the ratio \[ \boxed{2 : 7} \]