Question

Find the co-ordinates of the point dividing the line segment formed by joining the points (2, -3) and (-4, 6) in the ratio 1 : 2.

Hint:

To avoid errors, clearly label your points and ratio (\(x_1, y_1, x_2, y_2, m_1, m_2\)) before substituting them into the formula. This systematic approach minimizes the risk of mixing up values.

Answer 1

Step 1: Understanding the Concept:
This problem requires the use of the section formula, which gives the coordinates of a point that divides a line segment into a given ratio.
Step 2: Key Formula or Approach:
The coordinates \(P(x, y)\) of a point that divides the line segment joining \(A(x_1, y_1)\) and \(B(x_2, y_2)\) in the ratio \(m_1 : m_2\) are given by the section formula:
\[ P(x, y) = \left( \frac{m_1x_2 + m_2x_1}{m_1 + m_2}, \frac{m_1y_2 + m_2y_1}{m_1 + m_2} \right) \] Step 3: Detailed Explanation:
The given points are \( (2, -3) \) and \( (-4, 6) \). The ratio is \( 1 : 2 \).
Let \( (x_1, y_1) = (2, -3) \), \( (x_2, y_2) = (-4, 6) \), \( m_1 = 1 \), and \( m_2 = 2 \).
Now, substitute these values into the section formula.
For the x-coordinate:
\[ x = \frac{1(-4) + 2(2)}{1 + 2} = \frac{-4 + 4}{3} = \frac{0}{3} = 0 \] For the y-coordinate:
\[ y = \frac{1(6) + 2(-3)}{1 + 2} = \frac{6 - 6}{3} = \frac{0}{3} = 0 \] The coordinates of the point are (0, 0).
Step 4: Final Answer:
The co-ordinates of the point dividing the line segment are (0, 0), which is the origin.