Home
Class 12
MATHS
Let A,B,C,D,E represent vertices of a re...

Let A,B,C,D,E represent vertices of a regular pentangon ABCDE. Given the position vector of these vertices be a,a+b,b,`lamda a` and `lamdab` respectively.
Q. AD divides EC in the ratio

A

`cos""(2pi)/(5) : 1`

B

`cos""(3pi)/(5) : 1`

C

`1 : 2 cos""(2pi)/(5)`

D

`1 : 2`

Text Solution

Verified by Experts

The correct Answer is:
C
Given ABCDE is a regular pentagon

Let position vector of point A and C be `veca and veb`, respectively.
`AD` is parallel to BC and AB is parallel to EC.
Therefore,
AOCB is a parallelogram
and position vector of B is `veca + vecb`
The position vectors of E and D are `lamdavecb and lamda veca` respectively.
Also `OA = BC = AB = OC =1` (let)
Therefore, AOCB is rhombus.
`angle ABC = angle AOC = (3pi)/(5)`
and `angle OAB = angle BCO = pi - (3pi)/(5) = (2pi)/(5)`
Further `OA = AE = 1 and OC = CD = 1 `
Thus, `DeltaEAO and Delta OCD` are isosceles.
In `Delta OCD`, using sine rule we get `(OC)/(sin""(2pi)/(5))= (OD )/( sin""(pi)/(5))`
`rArr OD = (1)/( 2 cos ""(pi)/(5)) = OE`
`rArr AD = OA + OD = 1 + (1)/(2 cos ""(pi)/(2))`
`rArr (AD)/(BC) =1 + (1)/(2 cos""(pi)/(5)) = (1 +2 cos""(pi)/(5))/( 2 cos""(pi)/(5))`
And `(OE)/( OC) = (1)/( 2cos ""(pi)/(5))`
Promotional Banner

Topper's Solved these Questions

  • INTRODUCTION TO VECTORS

    CENGAGE ENGLISH|Exercise INTEGER TYPE|8 Videos

  • INTRODUCTION TO VECTORS

    CENGAGE ENGLISH|Exercise ARCHIVES SUBJECTIVE TYPE|9 Videos

  • INTRODUCTION TO VECTORS

    CENGAGE ENGLISH|Exercise REASONING TYPE|11 Videos

  • INTEGRALS

    CENGAGE ENGLISH|Exercise All Questions|764 Videos

  • INVERSE TRIGONOMETRIC FUNCTIONS

    CENGAGE ENGLISH|Exercise Archives (Numerical Value type)|2 Videos

Similar Questions

Explore conceptually related problems

If the position vectors of the points A,B and C be a,b and 3a-2b respectively, then prove that the points A,B and C are collinear.

veca,vecb,vecc are the position vectors of vertices A, B, C respectively of a paralleloram, ABCD, ifnd the position vector of D.

If a,b and c are the position vectors of the vertices A,B and C of the DeltaABC , then the centroid of DeltaABC is

Let A,B,C be vertices of a triangle ABC in which B is taken as origin of reference and position vectors of A and C are veca and vecc respectively. A line AR parallel to BC is drawn from A PR (P is the mid point of AB) meets AC and Q and area of triangle ACR is 2 times area of triangle ABC Positon vector of Q for position vector of R in (1) is (A) (2veca+3vecc)/5 (B) (3veca+2vecc)/5 (C) (veca+2vecc)/5 (D) none of these

Let A, B, C respresent the vertices of a triangle, where A is the origin and B and C have position b and c respectively. Points M, N and P are taken on sides AB, BC and CA respectively, such that (AM)/(AB)=(BN)/(BC)=(CP)/(CA)=alpha Q. AN+BP+CM is

Let A,B,C be vertices of a triangle ABC in which B is taken as origin of reference and position vectors of A and C are veca and vecc respectively. A line AR parallel to BC is drawn from A PR (P is the mid point of AB) meets AC and Q and area of triangle ACR is 2 times area of triangle ABC Position vector of R in terms veca and vecc is (A) veca+2vecc (B) veca+3vecc (C) veca+vecc (D) veca+4vecc

If vec a ,\ vec b ,\ vec c are position vectors of the vertices A ,\ B\ a n d\ C respectively, of a triangle A B C ,\ write the value of vec A B+ vec B C+ vec C Adot

A (-1, 0), B (1, 3) and D (3, 5) are the vertices of a parallelogram ABCD. Find the co-ordinates of vertex C.

In the pelvic girdle of man A,B,C,D and E respectively represent

If veca , vecb, vecc are the position vectors of the vertices. A,B,C of a triangle ABC. Then the area of triangle ABC is