Suppose A, B, C are defined as `A = a^(2)b + ab^(2) - a^(2)c - ac^(2), B = b^(2)c + bc^(2) - a^(2)b - ab^(2)`, and `C = a^(2)c + ac^(2) - b^(2)c - bc^(2)`, where `a gt b gt c gt 0` and the equation `Ax^(2) + Bx + C = 0` has equal roots, then a, b, c are in

Options

A. A.P.
B. G.P.
C. H.P.
D. A.G.P.

Answer

3

Explanation

A = a(b - c) (a + b + c)
B = b(c - a) ( a + b + c)
C = c(a - b) (a + b + c)
Now, `Ax^(2) + Bx + C = 0`
`rArr (a + b + c) {a(b - c)x^(2) + b(c - a)x + c(a - b) = 0`
Given that roots are equal. Hence,
D = 0
`rArr b^(2)(c - a)^(2) - 4ac(b - c)(a - b) = 0`
or `b^(2)c^(2) - 2ab^(2)c + b^(2)a^(2) - 4a^(2)bc + 4acb^(2) + 4a^(2)c^(2) - 4abc^(2) = 0`
or `b^(2)c^(2) + b^(2)a^(2) + 4a^(2)c^(2) + 2ab^(2)c - 4a^(2)bc - 4abc^(2) = 0`
or `(bc + ab - 2ac)^(2) = 0`
or `bc + ab = 2ac`
`rArr 1/a + 1/c = 2/b`
Hence, a, b, c are in H.P.