Math, asked by rjpchavhan1, 6 months ago

in a triangle Oab, e is the midpoint of bo and d is a point on ab such that ad:db =2:1 if od and ae intersect at p, determine the ratio op:pd using vector methods​

Answers

Answered by luvmanu301
0

Step-by-step explanation:

With O as origin let a and b be the position vectors of A and B respectively.

Then the position vector of E, the mid-point of OB, is

2

b

Again, since AD:DB=2:1, the position vector of D is

1+2

1⋅a+2b

=

3

a+2b

Equation of OD and AE are r=t

3

a+2b

...(1)

and r=a+s(

2

b

−a) or r=(1−s)a+s

2

b

...(2)

If they intersect at p, then we will have identical values of r.

Hence comparing the coefficients of a and b, we get

3

t

=1−s,

3

2t

=

2

s

∴t=

5

3

or s=

5

4

.

Putting for t in (1) or for s in (2), we get the position vector of point of intersection P as

5

a+2b

...(3)

Now let P divide OD in the ratio λ:1.

Hence by ratio formula the P.V. of P is

λ+1

3

λ(a+2b)

+1.0

=

3(λ+1)

λ

(a+2b) ....(4)

Comparing (3) and (4), we get

3(λ+1)

λ

=

5

1

⇒5λ=3λ+3⇒2λ=3⇒λ=

2

3

∴OP:PD=3:2

Similar questions