JEE Advance - Mathematics (1999 - No. 23)
Let a, b, c, d be real numbers in G.P. If u, v, w, satisfy the system of equations
u + 2v + 3w = 6
4u + 5v + 6w = 12
6u + 9v = 4
u + 2v + 3w = 6
4u + 5v + 6w = 12
6u + 9v = 4
then show that the roots of the equation $$\left( {{1 \over u} + {1 \over v} + {1 \over w}} \right){x^2}$$
$$ + [{(b - c)^2} + {(c - a)^2} + {(d - b)^2}]x + u + v + w = 0$$ and $$20{x^2} + 10{(a - d)^2}x - 9 = 0$$ are reciprocals of each other.
The roots of the first equation are always real and positive.
The roots of both equations are real, and one root of each equation is the reciprocal of a root of the other.
If a=b=c=d, the roots of the equations will be imaginary
The coefficients of the quadratic equations have no influence on the reciprocity of the roots.
The given system of equations is inconsistent.