Section formula

Jun 18, 2020, 16:45 IST

Section formula

Definition :-The section formula tells us the coordinates of the point which divides a given line segment into two parts such that their lengths are in the ratio m: n.

Formula description :-
Internal divisions with section formula

If point P(x, y) lies on line segment (between points A and B) and satisfies AP:PB = m : n, then we say that P divides internally in the ratio m : n. The point of division has the coordinates

External divisions with section formula :

If P = (x, y) lies on the extension of line segment(not lying between points A and B) and satisfies AP : PB = m : n, then we say that P dividesexternally in the ratio m : n. The point of division is

Application :-All the air traffic is controlled by air traffic controller. A controller must know the location of every aircraft at any particular instant of time in the sky. Coordinates of any particular vehicle are used to describe its current location of the aircraft. Even if an aircraft moves a small distance (up, down, forward or backward), its coordinates are updated in the system.

Example 1 :-Find the co-ordinates of point P which divides the line joining A = (4, –5) and B = (6, 3) in the ratio 2 : 5.

Solution :-Let the co-ordinates of P be (x, y).

Example 2 :-Z (4, 5) and x(7, – 1) are two given points and the point y divides the line-segment zx externally in the ratio 4:3. Find the coordinates of y.

Solution :-
Given that, z(4,5)=(x1,y1), x(7,-1)=(x2,y2)
Point y divides the segment zx in the ratio 4:3, hence m=4, n=3

Since it is mentioned in the question that the point y divides the segment externally we use the section formula for external division,

Formula: y={[(mx2-nx1)/(m-n)],[(my2-ny1)/(m-n)]}
Substituting the known values,
={[(4(7)-3(4))/(4-3)],[(4(-1)-3(5)/(4-3)]}
={(28-12)/1,(-4-15)/1} ={16,-19}
The coordinates for the point y are (16,-19)