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Vertices of a variable acute-angled triangle ABC lie on a circle of radius R such that da/dA + db/dB + dc/dC = 6. Distance of orthocentre of triangle ABC from vertices A, B and C is x1, x2 and x3, respectively

1. In-radius of triangle  ABC is 

(A)  1

(B)  2

(C)  3

(D)  4

2. Maximum value of x1x2x3 is 

(A)  4

(B)  6 

(C)  8 

(D)  10

3.  dx1/da + dx2/db + dx3/dc  is always less than equal to

(A)   -3√3

(B)  3√3

(c)  1

(D)  6

1 Answer

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Best answer

Correct option (C)(A)(D) 

1.  a = 2R sinA

da/dA = 2R cosA,db/dB = 2R cosB,dc/dC = 2R cosC

Now,

2.  x1 = 2R cosA, x2 = 2R cosB, x3 = 2R cosC 

⇒ x1 + x2 + x3 = 2R(cosA + cosB + cosC) = 6

Now,

3.  

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