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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 |
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Answer» `-SQRT3` `(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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