Latest Wipro Aptitude Question SOLUTION: If the sum of the roots of the equation ax2 + bx + c=0 is equal to the sum of the squares of their reciprocals then a/c, b/a, c/b are in  (a) AP (b) GP (c) HP (d) None of these

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If the sum of the roots of the equation ax2 + bx + c=0 is equal to the sum of the squares of their reciprocals then a/c, b/a, c/b are in 
(a) AP
(b) GP
(c) HP
(d) None of these

Asked In Wipro madhan (11 years ago)

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Kailash  (11 years ago)

Suppose the roots are r and s. Then (x-r)(x-s) = 0, so
x^2 - (r+s) x + (rs) = 0
and hence r+s = -b/a and rs = c/a.

We are saying that
r + s = (1/r^2 + 1/s^2)
(rs)^2 (r+s) = (rs)^2 (1/r^2 + 1/s^2) = s^2 + r^2
(rs)^2 (r+s) = (r+s)^2 - 2rs
(c/a)^2 (-b/a) = (-b/a)^2 - 2(c/a)
Multiplying through by a^3 we get
-bc^2 = ab^2 - 2a^2c
Divide through by abc to get
-c/a = b/c - 2a/b
or a/b - c/a = b/c - a/b
which means c/a, a/b and b/c are in AP.

Hence their reciprocals, a/c, b/a and c/b are in harmonic progression (HP).

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Shashank Mishra  (11 years ago)

in HP becoz c/a , a/b , b/c are in AP after solving equations
so answer is
(c)HP

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