If the point P(2,1) lies on the line segment joining the . . .

Question : If the point P(2,1) lies on the line segment joining the points A(4,2) and B(8,4) then 

a) AP=1/3 AB

b) AP=PB

c) PB=1/3AB

d) AP=1/2 AB

Doubt by Zoha [230821]

Similar Question 

If the point P(2,1) lies on the line segment joining the points A(4,2) and B(8,4), then …..

a) AP=1/3 AB

b) AP=PB

c) PB=1/3 AB

d) AP=1/2 AB

Doubt by Gauri [240921]

Solution : 

This questions can be solved by two methods

Method I

Let
A (4,2)

B (8,4)

P (2,1)

Let us consider that P divides AB in k:1

 ____________________

A (4,2)          P(2,1)           B(8,4)

Using Section Formula

P(x,y) = {[k(x₂)+1(x₁)]/(k+1), [k(y₂)+1(y₁)]/(k+1)}

P(2,1) = {[k(8)+1(4)]/(k+1), [k(4)+1(2)]/(k+1)}

Equating the x-coordinate both sides

2 = [k(8)+1(4)]/(k+1)
2 (k+1) = 8k+4
2k+2 = 8k+4
2k-8k = 4-2
-6k = 2
k = 2/(-6)
k = -1/3

k/1 = -1/3

∵ k is coming negative it means the the point P is not coming between the points A and B but it is coming outside of AB i.e. the point P divide the AB in 1:3 externally.

In order to divide AB in 1:3 the point P must lie on the left of A and not right of B as shown below

  __________________
P           A                     B

AP:BP = 1:3
AP/BP = 1/3
AP/(AP+AB) = 1/3 [∵BP = AP+AB]
3AP = AP+AB
3AP-AP = AB
2AP = AB
AP= 1/2 AB

Hence, d) is the correct option. 

Method II (Recommended)

Let
A (4,2)

B (8,4)

P (2,1)

By distance formula 

AB = √[(x₂-x₁)²+(y₂-y₁)²]
AB = √[(8-4)²+(4-2)²]
AB = √[16+4]
AB=√20

AB = √(2×2×5)
AB = 2√5 units

Similarly

PB = √[(8-2)²+(4-1)²]
PB = √[36+9]
PB = √45
PB = √(3×3×5)
PB = 3√5 units

AP = √[(4-2)²+(2-1)²]
AP = √[4+1]
AP=√5 units

Clearly 

AP/AB = √5/2√5
AP/AB = 1/2
AP = 1/2 AB

Hence, d) is the correct option.