Q13/069) given that a,b are roots of equation Ax^2 – 4x +1 = 0 and c , d are roots of equation Bx^2 – 6x + 1 = 0 find the values of A and B such that a,c, b,d are in HP.

a,c , b,d are in HP => 1/a, 1/c, 1/b and 1/d are in AP

A,b are roots of Ax^2 – 4x + 1 = 0 => 1/a, 1/b are roots of 1 -4x + A = 0

So 1/a + 1/b = 4

Similarly  1/c + 1/d =  6

As 1/c – 1/a = 1/d – 1/b we get 1/a = 1, 1/c = 2, 1/b = 3 and 1/d = 6.

using these we get A = 1/ac = 3 and B= 1/bd = 8