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if a,b,c are 3 distinct positive numbers in H.P. then the equation x^2 -kx + 2b^101-a^101-c^101=0, k belongs to R (A) has two distinct real roots. (B)has product of roots positive (C)has product of roots negative (D)has imaginary roots

if a,b,c are 3 distinct positive numbers in H.P. then the equation x^2 -kx + 2b^101-a^101-c^101=0, k belongs to R
(A) has two distinct real roots.
(B)has product of roots positive
(C)has product of roots negative
(D)has imaginary roots

Grade:11

1 Answers

SHAIK AASIF AHAMED
askIITians Faculty 74 Points
7 years ago
Hello student,
Please find the answer to your question below
We know that AM>GM
As a,b,c are in HP
(an+cn)/2>\sqrt{a^{n}c^{n}}…............(1)
But GM>HM
so\sqrt{ac}>b
or\sqrt{a^{n}c^{n}}>bn..................(2)
So from 1 and 2(an+cn)/2>\sqrt{a^{n}c^{n}}>bn
an+cn>2bn
S0 a101+c101-2b101>0 so2b^101-a^101-c^101<0...............(3)
So the product of roots have -ve sign
so C is the correct answer

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