Question

b) OABC is a parallelogram. P and Q divide OC and BC in the ratio
CP : PO CQ: QB = 1 : 3. If OÁ = à and OČ= =, find the vector PQ and show
that PQ // OB.
​

Answer 1

Step-by-step explanation:

Let the position vector of A and C be

a

and

c

respectively. Therefore,

Position vector of B=

b

=

a

+

c

...........................................................(i)

Also Position vector of E=

3

b

+2

c

=

3

a

+3

c

............................................................(ii)

Now point P lies on angle bisector of ∠AOC. Thus,

Position vector of point P=λ(

∣

a

∣

a

+

∣

b

∣

b

)............................................................................(iii)

Also let P divides EA in ration μ:1. Therefore,

Position vector of P

=

μ+1

μ

a

+

3

a

+3

c

=

3(μ+1)

(3μ+1)

a

+3

c

................................................................................(iv)

Comparing (iii) and (iv), we get

λ(

∣

a

∣

a

+

∣

c

∣

c

)=

3(μ+1)

(3μ+1)

a

+3

c

⇒

∣

a

∣

λ

=

3(μ+1)

3μ+1

and

∣

c

∣

λ

=

μ+1

1

⇒

3∣

a

∣

3∣

c

∣−∣

a

∣

=μ

⇒

∣

c

∣

λ

=

3∣

a

∣

3∣

c

∣−

a

+1

1

⇒λ=

3∣

c

∣+2∣

a

∣

3∣

a

∣∣

c

∣

Hence, position vector of P is

3∣

c

∣+2∣

c

∣

3∣

a

∣∣

c

∣

(

∣

a

∣

a

+

∣

c

∣

c

)

Let F divides AB in ratio t:1, then position vector of F is

t+1

t

b

+

a

solution