InterviewSolution
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InterviewSolution
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Let `a ,b ,c ,d`be four distinct real numbers in A.P. Then half of the smallestpositive valueof `k`satisfying `a(a-b)+k(b-c)^2=(c-a)^3=2(a-x)+(b-d)^2+(c-d)^3`is __________. |
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Answer» Correct Answer - 16 Since a,b,c,d are in A.P., we have b-a=c-b=d-c=D (let common difference) or d=a+3D `rArra-d=-3D and d=b+2D` or b-d=-2D Also c=a+2D or c-a=2D `therefore` Given equation 2(a-b)`+k(b-c)^(2)+(c-a)^(3)` `=2(a-d)+(b-d)^(2)+(c-d)^(3)` becomes `-2D+kD^(2)+(2D)^(3)=-6D+4D^(2)-D^(3)` `rArr9D^(2)+(k-4)D+4=0` Since D is real, we have `(k-4)^(2)-4(4)(9)ge0` or `k^(2)-8k-128ge0` or `(k-16)(k+8)ge0` `thereforekin(-oo,-8]cup[16,oo)` Hence, the smallest positive value of k = 16. |
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