Home
Class 12
MATHS
P(cosalpha,sinalpha), Q(cosbeta, sinbeta...

`P(cosalpha,sinalpha), Q(cosbeta, sinbeta) , R(cosgamma, singamma)` are vertices of triangle whose orthocenter is `(0, 0)` then the value of `cos(alpha-beta) + cos(beta-gamma) + cos(gamma-alpha)` is

A

`-3//2`

B

`-1//2`

C

`½`

D

`3//2`

Text Solution

Verified by Experts

The correct Answer is:
A
Clearly `OP = OQ = OR`, where O is orthocentre
`:. O(0,0)` is the circum centre of `DeltaPQR`.
Also, given orthocenter `= (0,0)`
Thus, orthocentre and circumcentre coincide, So, triangle is equilateral
`:.` Centroid of `DeltaPQR = (0,0)`
`rArr cos alpha +cos beta +cos gamma =0`, and `sin alpha + sin beta+ sin gamma = 0` Squaring and adding we get
`cos(alpha-beta) +cos (beta-gamma) +cos (gamma- alpha) =-(3)/(2)`
Promotional Banner

Topper's Solved these Questions

  • COORDINATE SYSTEM

    CENGAGE ENGLISH|Exercise Comprehension Type|4 Videos

  • COORDINATE SYSTEM

    CENGAGE ENGLISH|Exercise Multiple Correct Answers Type|2 Videos

  • CONTINUITY AND DIFFERENTIABILITY

    CENGAGE ENGLISH|Exercise Comprehension Type|2 Videos

  • COORDINATE SYSYEM

    CENGAGE ENGLISH|Exercise JEE Main|6 Videos

Similar Questions

Explore conceptually related problems

If A(cos alpha,sin alpha), B(cos beta, sin beta), C(cosgamma,sin gamma) are vertices of Delta ABC having ortho centre at origin then find the value of sumcos(alpha-beta).

The value of cos^(2) alpha +cos^(2) beta +cos^(2) gamma is _____ .

If cosalpha, cosbeta and cosgamma are the direction cosine of a line, then find the value of cos^2alpha+(cosbeta+singamma)(cosbeta-singamma) .

If alpha,beta,gamma are roots of the equation x^(3)+px+q=0 then the value of |{:(alpha,beta,gamma),(beta,gamma,alpha),(gamma,alpha,beta):}| is

If alpha, beta, gamma are the roots of x^(3) + ax^(2) + b = 0 , then the value of |(alpha,beta,gamma),(beta,gamma,alpha),(gamma,alpha,beta)| , is

If sinalpha-sinbeta=a and cosalpha+cosbeta=b then write the value of "cos"(alpha+beta) .

Prove that : cos^2 (beta-gamma) + cos^2 (gamma-alpha) + cos^2 (alpha-beta) =1+2cos (beta-gamma) cos (gamma-alpha) cos (alpha-beta) .

Let alpha, beta, gamma be the measures of angle such that sin alpha+sin beta+sin gamma ge 2 . Then the possible value of cos alpha + cos beta+cos gamma is

f(alpha,beta) = cos^2(alpha)+ cos^2(alpha+beta)- 2 cosalpha cosbeta cos(alpha+beta) is

If |{:(1,cos alpha, cos beta),(cos alpha, 1 , cos gamma ),(cos beta, cos gamma , 1):}|=|{:(0,cos alpha, cos beta),(cos alpha , 0 , cos gamma),(cos beta, cos gamma, 0):}| then the value of cos^2 alpha + cos^2 beta + cos^2 gamma is : (a) 1 (b) 1/2 (c) 3/8 (d) 9/4