Section Formula for External Division | Coordinate Geometry

Quick answer: A point $P$ divides the segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ externally in the ratio $m : n$ if $P$ lies on line $AB$ outside the segment and $\frac{AP}{BP} = \frac{m}{n}$. Its coordinates are $P = \left(\frac{mx_2 - nx_1}{m - n}, \frac{my_2 - ny_1}{m - n}\right)$, the internal section formula with $n$ replaced by $-n$. For example, the point dividing $A(2, 1)$ and $B(4, 3)$ externally in the ratio $3 : 2$ is $P(8, 7)$.

The section formula in Class 10 finds a point that divides a line segment internally, i.e. a point between the two end points. A point can also lie on the same line but outside the segment, beyond one of the end points. Such a point divides the segment externally.

Syllabus note: The CBSE Class 10 NCERT textbook (Chapter 7, Coordinate Geometry) covers only internal division. External division is taught in higher classes. This article is enrichment material.

What You Will Learn

• The difference between internal and external division
• The formula for external division and where it comes from
• How a negative value of $k$ in the ratio $k : 1$ shows external division
• A solved example, checked with the distance formula

1. Internal vs External Division

 Internal:  A --------- P --------- B
            (P is between A and B)

 External:  A --------------- B ---- P
            (P is beyond B)

2. The External Section Formula

Let $P(x, y)$ divide the segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ externally in the ratio $m : n$.

Step 1: Use the ratio along the x-direction. Since $A$, $B$ and $P$ lie on one line, the horizontal distances are in the same ratio as the lengths:

$\frac{x - x_1}{x - x_2} = \frac{m}{n}$

Step 2: Cross-multiply.

$n(x - x_1) = m(x - x_2)$

$nx - nx_1 = mx - mx_2$

Step 3: Solve for $x$.

$mx_2 - nx_1 = (m - n)x \implies x = \frac{mx_2 - nx_1}{m - n}$

Step 4: Do the same for $y$.

$y = \frac{my_2 - ny_1}{m - n}$

External section formula: $P = \left(\frac{mx_2 - nx_1}{m - n},\ \frac{my_2 - ny_1}{m - n}\right)$

Shortcut: this is the internal section formula $\left(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\right)$ with $n$ replaced by $-n$.

3. What a Negative k Means

In many problems we take the unknown ratio as $k : 1$ and use the internal section formula:

$x = \frac{kx_2 + x_1}{k + 1} \qquad y = \frac{ky_2 + y_1}{k + 1}$

For example, $k = -\frac{2}{3}$ means external division in the ratio $2 : 3$.

4. Solved Example

Question

Find the point $P$ that divides the segment joining $A(2, 1)$ and $B(4, 3)$ externally in the ratio $3 : 2$.

Solution

Step 1: Write the values. $x_1 = 2$, $y_1 = 1$, $x_2 = 4$, $y_2 = 3$, $m = 3$, $n = 2$, so $m - n = 1$.

Step 2: Find $x$.

$x = \frac{3(4) - 2(2)}{3 - 2} = \frac{12 - 4}{1} = 8$

Step 3: Find $y$.

$y = \frac{3(3) - 2(1)}{3 - 2} = \frac{9 - 2}{1} = 7$

Answer: $P = (8, 7)$.

Check using the distance formula:

• $AP = \sqrt{(8 - 2)^2 + (7 - 1)^2} = \sqrt{72} = 6\sqrt{2}$
• $BP = \sqrt{(8 - 4)^2 + (7 - 3)^2} = \sqrt{32} = 4\sqrt{2}$
• $\frac{AP}{BP} = \frac{6\sqrt{2}}{4\sqrt{2}} = \frac{3}{2}$, which matches the given ratio $3 : 2$. ✓

5. Summary Table

6. Tips and Common Mistakes

Tip: Remember the external formula as the internal formula with $n$ changed to $-n$.

Tip: Check your answer with the distance formula: $\frac{AP}{BP}$ should equal $\frac{m}{n}$.

Common Mistake: Writing $mx_1 - nx_2$. In both formulas, $m$ multiplies the coordinates of the second point, $B$, and $n$ those of the first point, $A$.

Common Mistake: Using the external formula with $m = n$. A point cannot divide a segment externally in the ratio $1 : 1$, because the denominator $m - n$ would be zero.

Frequently Asked Questions

What is external division of a line segment?

A point P divides AB externally in the ratio m : n if P lies on line AB outside the segment and AP : BP = m : n.

What is the formula for external division?

P = ((mx₂ − nx₁)/(m − n), (my₂ − ny₁)/(m − n)).

What does a negative value of k mean in the section formula?

If you take the ratio as k : 1 and get k < 0, the point divides the segment externally in the ratio |k| : 1.

Is external division in the Class 10 syllabus?

No. The Class 10 NCERT textbook covers only internal division; external division is taught in higher classes.