1.

Vertices of a variable acute angled triangle ABC lies on a fixed circle. Also a, b, c andA, B, C are lengths of sides and angles of triangle ABC, respectively. If x_(1),x_(2) and x_(3) are distances of orthocentre from A, B and C, respectively, then the maximum value of ((dx_(1))/(da)+(dx_(2))/(db)+(dx_(3))/(dc)) is

Answer»

`-SQRT3`
`-3SQRT3`
`sqrt3`
`3sqrt3`

Solution :`x_(1)=2R COS A, x_(2)=2R cos B,x_(3)=2R cos C`
`(dx_(1))/(dA)=-2R sin A`
Also `a=2R sin A rArr (da)/(dA)=2R cos A`
`"So,"(dx_(1))/(da)=-tanA,(dx_(2))/(db)=-tanB,(dx_(3))/(dc)=-tanC`
`"Now"tanA+tanB+tanCge3sqrt3`
`"So,"((dx_(1))/(da)+(dx_(2))/(db)+(dx_(3))/(dc))lt-3sqrt3`


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