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 verify your ratio with the y-coordinate: $\frac{2(-8) + 7(10)}{2+7} = \frac{-16+70}{9} = \frac{54}{9} = 6$. It matches!

Answer 1

Step 1: Understanding the Concept:
We use the section formula. If a point $P(x, y)$ divides a line segment $AB$ internally in the ratio $k:1$, its coordinates are given by: \[ x = \frac{kx_2 + x_1}{k+1}, \quad y = \frac{ky_2 + y_1}{k+1} \]
Step 2: Setting up the Equation:
Let the ratio be $k:1$.
Given: $P(-4, 6)$, $A(-6, 10)$, $B(3, -8)$.
Using the x-coordinate: \[ -4 = \frac{k(3) + (-6)}{k+1} \]
Step 3: Solving for k:
\[ -4(k + 1) = 3k - 6 \] \[ -4k - 4 = 3k - 6 \] \[ -4k - 3k = -6 + 4 \] \[ -7k = -2 \implies k = \frac{2}{7} \]
Step 4: Final Answer:
The ratio $k:1$ is $2/7:1$, which is 2:7.