MP Board · Class 10 · Mathematics · Coordinate GeometryFind the coordinates of the point which divides the join of $(-1, 7)$ and $(4, -3)$ in the ratio $2:3$ internally. Also, explain the section formula concept in detail.

Introduction to the Section Formula\nIn coordinate geometry, the section formula is a powerful tool utilized to find the coordinates of a point that divides a given line segment into two parts in a specific given ratio. This concept bridges the geometric understanding of ratios and proportions with algebraic coordinate representation.

Detailed Explanation of the Section Formula Concept

1. Definition of Internal Division: When a point $P(x, y)$ lies on the line segment joining two distinct points $A(x_1, y_1)$ and $B(x_2, y_2)$ such that it divides the segment into two parts whose lengths are in the ratio $m_1 : m_2$, it is called internal division.
2. Derivation Overview (Conceptual): By dropping perpendiculars from points $A, P,$ and $B$ onto the coordinate axes and using the properties of similar triangles (specifically through intercept theorems), the exact algebraic coordinates are derived.
3. The Section Formula Statement: If a point $(x, y)$ divides the line segment joining the points $(x_1, y_1)$ and $(x_2, y_2)$ internally in the ratio $m_1 : m_2$, then the coordinates of the point are given by: $$x = \frac{m_1x_2 + m_2x_1}{m_1 + m_2}$$ $$y = \frac{m_1y_2 + m_2y_1}{m_1 + m_2}$$
4. Special Case (Midpoint Formula): If the ratio is $1:1$ (meaning the point is the exact midpoint), the formula simplifies to: $$x = \frac{x_1 + x_2}{2}, \quad y = \frac{y_1 + y_2}{2}$$

Numerical Step-by-Step Solution\nNow, let us find the coordinates of the point dividing the line segment joining $(-1, 7)$ and $(4, -3)$ in the ratio $2:3$.

• Step 1: Identify the given values:

  • Point $A(x_1, y_1) = (-1, 7)$
  • Point $B(x_2, y_2) = (4, -3)$
  • Ratio $m_1 : m_2 = 2 : 3$

• Step 2: Apply the section formula for the $x$-coordinate: $$x = \frac{m_1x_2 + m_2x_1}{m_1 + m_2}$$ $$x = \frac{(2)(4) + (3)(-1)}{2 + 3}$$ $$x = \frac{8 - 3}{5}$$ $$x = \frac{5}{5} = 1$$

• Step 3: Apply the section formula for the $y$-coordinate: $$y = \frac{m_1y_2 + m_2y_1}{m_1 + m_2}$$ $$y = \frac{(2)(-3) + (3)(7)}{2 + 3}$$ $$y = \frac{-6 + 21}{5}$$ $$y = \frac{15}{5} = 3$$

Conclusion\nThe coordinates of the point that divides the given line segment in the ratio $2:3$ internally are $(1, 3)$.