Coordinate Geometry 07A: external division of a line - PART (1)

A point D that lie on the same line, however outside the join of two known points A and B, then that point would be called to external divide the AB line segment

Let's go deeply into that:

Consider a line segment AB and on the same line that joins A and B points, there is another point D which lies on the same line, just externally to the line segment AB.

Now let's go for a critical analysis of the situation given above:

(1) Point D lies externally to AB line segment

(2) Point D, say that, divides the join of A towards B externally in ratio m : n

(3) this same situation can also be seen from the concept of internal division.

We say that the join of A to D is divided internally by point B. Let's have a look on how our situation now becomes

The point B now divides AD internally in the ratio (m - n) : n

Now we could proceed the situation easily in terms of internal division.

Calculation

We have re-established the external division in terms of internal division. That's a big win..

But our aim is the same.. here is our aim.
Aim: We know coordinates of point A as $(x_1, y_1)$ and that of point B are also known. Let say B lies on ($x_2, y_2$) on the coordinate plane.

But for point D (point D lies on the same line join, just outside AB segment), it's coordinates are not known to us.
A--> known $(x_1, y_1)$
B-->known $(x_2, y_2)$
D--> to calculate $(\alpha, \beta)$

Now we could classify the same condition into two ways :

Firstly, we could say join of A towards D is divided internally by point B... This is simple in thinking, but since D is unknown we should go for a second line of thought.

Internal division Approach

Secondly, we could say that join of A towards B is divided externally by point D.

This is a better approach.

Here, is the formula that we shall apply

 $\alpha=\frac{mx_2 - nx_1}{m-n}$; $\beta=\frac{my_2 - ny_1}{m-n}$

External division approach

A-->$(x_1,y_1)$

B->$(x_2,y_2)$

D-->$(\alpha,\beta)$

It's never good to just mug up formula, without knowing how we get that. However, I believe that our readers deserve a break in reading this so we should rather take that in next post as part (2).

In this part 1, I meant to critically explain you the scenario... It’s always good to know the scenario.