Question

In which ratio, does the point (–4, 6) divides the line segment made by joining the points A(–6, 10) and B(3, –8)?

Hint:

When asked to find the ratio, assuming the ratio as \(k:1\) is generally faster than assuming \(m:n\). It reduces the problem to solving for a single variable, \(k\).

Answer 1

Step 1: Understanding the Concept:
We need to find the ratio in which a given point divides the line segment connecting two other points. This can be solved using the section formula.
Step 2: Key Formula or Approach:
Let the point P(x, y) divide the line segment joining A(\(x_1, y_1\)) and B(\(x_2, y_2\)) in the ratio \(k : 1\). The coordinates of P are given by:
\[ P(x, y) = \left( \frac{kx_2 + x_1}{k + 1}, \frac{ky_2 + y_1}{k + 1} \right) \] Step 3: Detailed Explanation:
Here, the dividing point is P(–4, 6), and the endpoints are A(–6, 10) and B(3, –8).
So, \( (x, y) = (–4, 6) \), \( (x_1, y_1) = (–6, 10) \), and \( (x_2, y_2) = (3, –8) \).
Let the required ratio be \(k : 1\).
Using the section formula for the x-coordinate:
\[ x = \frac{kx_2 + x_1}{k + 1} \] \[ -4 = \frac{k(3) + (-6)}{k + 1} \] \[ -4(k + 1) = 3k - 6 \] \[ -4k - 4 = 3k - 6 \] \[ 6 - 4 = 3k + 4k \] \[ 2 = 7k \] \[ k = \frac{2}{7} \] The ratio is \(k : 1\), which is \( \frac{2}{7} : 1 \). Multiplying by 7, we get the ratio 2 : 7.
We can verify this using the y-coordinate:
\[ y = \frac{ky_2 + y_1}{k + 1} \] \[ 6 = \frac{k(-8) + 10}{k + 1} \] \[ 6(k + 1) = -8k + 10 \] \[ 6k + 6 = -8k + 10 \] \[ 6k + 8k = 10 - 6 \] \[ 14k = 4 \] \[ k = \frac{4}{14} = \frac{2}{7} \] Both coordinates give the same value of \(k\), confirming our result.
Step 4: Final Answer:
The point (–4, 6) divides the line segment in the ratio 2 : 7.