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ABCD is a parallelogram. L is a point on...

ABCD is a parallelogram. L is a point on BC which divides BC in the ratio `1:2`. AL intersects BD at P.M is a point on DC which divides DC in the ratio `1 : 2` and AM intersects BD in Q.
Point Q divides DB in the ratio

A

`1:2`

B

`1:3`

C

`3:1`

D

`2:1`

Text Solution

Verified by Experts

The correct Answer is:
B

`vec(BL) = (1)/(3) vecb`
`therefore vec(AL) = veca + (1)/(3)vecb`
Let `vec(AP) = lamda ve(AL) and P` divides DB in the ratio `mu : 1 - mu`. Then
`" " vec(AP) =lamda veca + (lamda )/(3) vecb" "` (i)
Also `vec(AP) = mu veca + (1-mu) vecb)" "`(ii)
From (i) and (ii), `lamda veca + (lamda )/(3 ) vecb = mu veca + (1- mu) vecb`
` therefore " " lamda = mu and (lamda)/(3) = 1- mu`
`therefore " " lamda = (3)/(4)`
Hence, P divides AL in the ratio `3 : 1` and P divides DB in the ratio `3 :1`.
Similarly, Q divides DB in the ratio `1:3`.
Thus, `DQ = (1)/(4) DB and PB = (1)/(4) DB`
`therefore " " PQ = (1)/(2) DB`, i.e., `PQ : DB= 1 : 2`
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