Question

19. If the point P (2,1) lies on the line segment joining points A (4,2) and B (8,4), then
(a) AP = 1/3 AB (b) AP = BP (c) PB = 1/3 AB (d) AP = 1/2 AB​

Answer 1

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