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If the point P (2, 1) lies on the segment joining Points A (4, 2) and B (8, 4) then
Using the section formula, if a point (x,y) divides the line joining the points (x
1
,y
1
) and (x
2
,y
2
) in the ratio m:n, then
(x,y)=(
m+n
mx
2
+nx
1
,
m+n
my
2
+ny
1
)
Let P divide AB in the ratio k:1
Substituting (x
1
,y
1
)=(4,2) and (x
2
,y
2
)=(8,4) in the section formula, we get
P=(
k+1
k(8)+1(4)
,
k+1
k(4)+1(2)
)
But given P=(2,1)
⇒(
k+1
8k+4
,
k+1
4k+2
)=(2,1)
Comparing the x - coordinate,
⇒
k+1
8k+4
=2
⇒8k+4=2k+2
6k=−2
k=−
3
1
As k is negative, P divides AB in the ratio 1:3 externally.
⇒
PB
AP
=
3
1
⇒AP=
2
1
AB
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