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Through the point P(alpha,beta) , where ...

Through the point `P(alpha,beta)` , where `alphabeta>0,` the straight line `x/a+y/b=1` is drawn so as to form a triangle of area `S` with the axes. If `a b >0,` then the least value of `S` is `alphabeta` (b) `2alphabeta` (c) `3alphabeta` (d) none

A

`alphabeta`

B

`2alphabeta`

C

`3alphabeta`

D

none

Text Solution

Verified by Experts

The correct Answer is:
B
`OA=OQ+QA=alpha+betacottheta`
`OB=OR+RB=beta+alphatantheta`
`(1)/(2)(alpha+betacot)(beta+alphatantheta)`
`(1)/(2)(2alphabeta+alpha^2tantheta+beta^2cottheta)`
`(1)/(2)(2alphabeta+(alphasqrttantheta-betasqrtcottheta)^2+2alphabeta)`
`ge(1)/(2)(2alphabeta+2alphabeta)`
`=2alphabeta`
Least value of `S=2alphabeta`.
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