Let a , b be two non-zero real numbers. If p and r are the roots of the equation x 2 - 8 a x + 2 a = 0 and q and s are the roots of the equation x 2 + 12 b x + 6 b = 0 , such that 1 p , 1 q , 1 r , 1 s are in A.P., then a - 1 - b - 1 is equal to _____ .

Step-by-step solution

Since p ,   r are roots of x 2 - 8 a x + 2 a = 0 and q , s are the roots of x 2 + 12 b x + 6 b = 0 So the equation 2 a x 2 - 8 a x + 1 = 0 has roots 1 p , 1 r and 6 b x 2 + 12 b x + 1 = 0 has roots 1 q , 1 s . Given that 1 p , 1 q , 1 r , 1 s are in A.P. Let 1 p = α - 3 β , 1 q = α - β , 1 r = α + β and 1 s = α + 3 β i.e. 1 p + 1 r = 2 α - 2 β = 4 and 1 q + 1 s = 2 α + 2 β = - 2 or α = 1 2 ,   β = - 3 2 i.e. 1 p = 5 ,   1 q = 2 , &