Coordinate Geometry of Class 10
FORMULA FOR INTERNAL DIVISION:
The coordinates of the pint which divided the line segment joining the pints (x1, y1) and x2, y2) internally in the ratio m : n
are given by
Proof: Let O be the origin and let OX and OY be the X-axis and Y-axis respectively. Let A(x1, y1) and B(x2, y2) bet the given points. Let (x, y) be the coordinates of the point p which divides AB internally in the ratio m : n Draw ALOX, BMOX, PNOX. Also draw AH and PK perpendicular from A and P on PN and BM respectively. Then
OL = x1, ON = x, OM = x2, AL = y1, PN = y and BM = y2.
∴ AH = LN = ON – OL = x – x1, PH = PH – HN
= PN – AL = y – y1, PK = NM = OM – ON = x2 - x
and BK = BM – MK = BM – PN = y2 – y.
Clearly, ΔAHP and ΔPKB are similar.
⇒ mx2 – mx = nx – nx1 ⇒ mx + nx = mx2 + nx1
⇒ my2 – my = ny – ny1 ⇒ my + ny = my2 + ny1
Thus the coordinates of P are
FORMULA FOR EXTERNAL DIVISION:
The coordinates of the points which divides the line segment joining the points (x1, y1) and (x2, y2) externally in the ratio m : n are given by