Section Formulae in Coordinate Geometry

In coordinate geometry, the section formulae are used to find the coordinates of a point that divides a line segment into two parts in a given ratio. There are two main types of section formulae: the internal division formula and the external division formula.

Internal Division Formula

The internal division formula is used when a point divides a line segment joining two points internally in a given ratio.

Formula

If a line segment with endpoints $A(x_1, y_1)$ and $B(x_2, y_2)$ is divided by a point $P(x, y)$ in the ratio $m:n$, then the coordinates of point $P$ are given by:

$$ x = \frac{mx_2 + nx_1}{m + n}, \quad y = \frac{my_2 + ny_1}{m + n} $$

Example

Let's find the coordinates of the point which divides the line segment joining the points $A(2, 3)$ and $B(8, 5)$ in the ratio $2:3$.

Using the internal division formula:

$$ x = \frac{2 \cdot 8 + 3 \cdot 2}{2 + 3} = \frac{16 + 6}{5} = \frac{22}{5} = 4.4 $$

$$ y = \frac{2 \cdot 5 + 3 \cdot 3}{2 + 3} = \frac{10 + 9}{5} = \frac{19}{5} = 3.8 $$

So, the coordinates of the point are $(4.4, 3.8)$.

External Division Formula

The external division formula is used when a point divides a line segment joining two points externally in a given ratio.

Formula

If a line segment with endpoints $A(x_1, y_1)$ and $B(x_2, y_2)$ is divided by a point $P(x, y)$ externally in the ratio $m:n$, then the coordinates of point $P$ are given by:

$$ x = \frac{mx_2 - nx_1}{m - n}, \quad y = \frac{my_2 - ny_1}{m - n} $$

Example

Let's find the coordinates of the point which divides the line segment joining the points $A(2, 3)$ and $B(8, 5)$ externally in the ratio $2:3$.

Using the external division formula:

$$ x = \frac{2 \cdot 8 - 3 \cdot 2}{2 - 3} = \frac{16 - 6}{-1} = -10 $$

$$ y = \frac{2 \cdot 5 - 3 \cdot 3}{2 - 3} = \frac{10 - 9}{-1} = -1 $$

So, the coordinates of the point are $(-10, -1)$.

Comparison Table

Here's a comparison table highlighting the differences between internal and external division formulae:

Feature	Internal Division	External Division
Type of Division	Divides line segment internally	Divides line segment externally
Ratio	Positive ratio $m:n$	Positive ratio $m:n$
Formula for $x$	$\frac{mx_2 + nx_1}{m + n}$	$\frac{mx_2 - nx_1}{m - n}$
Formula for $y$	$\frac{my_2 + ny_1}{m + n}$	$\frac{my_2 - ny_1}{m - n}$
Sum or Difference in Denominator	Sum ($m + n$)	Difference ($m - n$)

Important Points

• The section formulae are applicable in both two-dimensional and three-dimensional geometry.
• In the case of internal division, the point lies between the endpoints of the line segment.
• In the case of external division, the point does not lie between the endpoints and is located outside the line segment.
• If the ratio is $1:1$ in the internal division formula, the point is the midpoint of the line segment, and the formula simplifies to the midpoint formula: $x = \frac{x_1 + x_2}{2}, y = \frac{y_1 + y_2}{2}$.
• The section formulae can be derived using the concept of similar triangles or vector algebra.

Understanding and applying these formulae are crucial for solving problems in coordinate geometry, especially in competitive exams where questions on this topic are common.