Home
Class 12
MATHS
Vertices of a variable acute angled tria...

Vertices of a variable acute angled triangle ABC lies on a fixed circle. Also a, b, c and A, 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

A

`-sqrt3`

B

`-3sqrt3`

C

`sqrt3`

D

`3sqrt3`

Text Solution

Verified by Experts

The correct Answer is:
B
`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`
Promotional Banner

Topper's Solved these Questions

  • METHODS OF DIFFERENTIATION

    CENGAGE|Exercise Multiple Correct Answer Type|7 Videos

  • MATRICES

    CENGAGE|Exercise Multiple Correct Answer|7 Videos

  • METHODS OF DIFFERETIATION

    CENGAGE|Exercise Question Bank|13 Videos

Similar Questions

Explore conceptually related problems

Vertices of a variable acute angled triangle A B C lies on a fixed circle. Also, a ,b ,ca n dA ,B ,C are lengths of sides and angles of triangle A B C , respectively. If x_1, x_2a n dx_3 are distances of orthocenter from A ,Ba n dC , respectively, then the maximum value of ((dx_1)/(d a)+(dx_2)/(d b)+(dx_3)/(d c)) is -sqrt(3) b. -3sqrt(3) c. sqrt(3) d. 3sqrt(3)

If circumradius of triangle is 2, then the maximum value of (abc)/(a+b+c) is

In triangle ABC, angle B is right angled, AC=2 and A(2,2), B(1,3) then the length of the median AD is

If in triangle ABC, a, c and angle A are given and c sin A lt a lt c , then ( b_(1) and b_(2) are values of b)

If angle C of triangle ABC is 90^0, then prove that tanA+tanB=(c^2)/(a b) (where, a , b , c , are sides opposite to angles A , B , C , respectively).

If x , ya n dz are the distances of incenter from the vertices of the triangle A B C , respectively, then prove that (a b c)/(x y z)=cot(A/2)cot(B/2)cot(C/2)

If A , B and C are interior angles of a triangle ABC, then show that tan ((A+B) /(2)) =cot ©/(2)

If the angles A, B and C of a triangle are in an arithmetic progression and if a, b and c denote the lengths of the sides opposite to A, B and C respectively, then the value of the expression (a)/(c) sin 2C + (c)/(a) sin 2A is

If A , B and C are interior angles of triangle ABC ,then show that sin ((B+C)/( 2) )=cos (A) /(2)