Section Formula (External Division)

In coordinate geometry, the section formula is a mathematical tool used to find the coordinates of a point that divides a line segment externally in a given ratio. This is particularly useful when we have a line segment with endpoints and we need to find a point outside the line segment that divides it into a particular ratio.

Understanding External Division

When we say a point divides a line segment externally, we mean that the point lies outside the line segment and not between the endpoints. The external division of a line segment is based on the concept of directed line segments, where the ratio in which the point divides the line segment can be positive or negative.

Section Formula for External Division

Let's consider a line segment with endpoints $A(x_1, y_1)$ and $B(x_2, y_2)$. Suppose a point $P(x, y)$ divides the line segment $AB$ externally in the ratio $m:n$. The coordinates of point $P$ can be determined using the section formula for external division as follows:

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

Differences and Important Points

Here's a table summarizing the differences between internal and external division:

Aspect	Internal Division	External Division
Position of Divider	Between endpoints	Outside the endpoints
Ratio	Positive	Can be positive or negative
Formula	$x = \frac{mx_2 + nx_1}{m + n}$	$x = \frac{mx_2 - nx_1}{m - n}$
	$y = \frac{my_2 + ny_1}{m + n}$	$y = \frac{my_2 - ny_1}{m - n}$
Nature of Division	Segment is divided into two parts	Segment is extended and point lies beyond

Examples

Example 1: Finding a Point Dividing a Line Segment Externally

Given a line segment with endpoints $A(2, 3)$ and $B(4, -1)$, find the coordinates of a point $P$ that divides the line segment $AB$ externally in the ratio $2:3$.

Using the section formula for external division:

$$ x = \frac{2 \cdot 4 - 3 \cdot 2}{2 - 3} = \frac{8 - 6}{-1} = \frac{2}{-1} = -2 $$

$$ y = \frac{2 \cdot (-1) - 3 \cdot 3}{2 - 3} = \frac{-2 - 9}{-1} = \frac{-11}{-1} = 11 $$

Therefore, the coordinates of point $P$ are $(-2, 11)$.

Example 2: Verifying the External Division

Let's verify the result of Example 1 by checking if the distances from $A$ to $P$ and $B$ to $P$ are in the ratio $2:3$.

Distance $AP$:

$$ AP = \sqrt{(-2 - 2)^2 + (11 - 3)^2} = \sqrt{(-4)^2 + (8)^2} = \sqrt{16 + 64} = \sqrt{80} $$

Distance $BP$:

$$ BP = \sqrt{(-2 - 4)^2 + (11 + 1)^2} = \sqrt{(-6)^2 + (12)^2} = \sqrt{36 + 144} = \sqrt{180} $$

Now, we check the ratio:

$$ \frac{AP}{BP} = \frac{\sqrt{80}}{\sqrt{180}} = \frac{\sqrt{4 \cdot 20}}{\sqrt{9 \cdot 20}} = \frac{2\sqrt{20}}{3\sqrt{20}} = \frac{2}{3} $$

The distances $AP$ and $BP$ are indeed in the ratio $2:3$, which verifies our result.

Conclusion

The section formula for external division is a powerful tool in coordinate geometry for finding the coordinates of a point that divides a line segment externally in a given ratio. Understanding the differences between internal and external division is crucial for applying the correct formula and solving problems accurately.