A homogeneous system of linear equations has a unique solution (the trivial solution) if and only if its determinant is non-zero. If this determinant is zero, then the system has an infinite number of solutions.
So, the system of equations
has a non trivial solution, it means that :
We know that
Since , So,
this means that either
cannot be zero because:
As , The given value cannot be zero.
This means (a + b + c ) has to be zero.
So, , and , so,
The quadratic equation has real solutions iff
Since, a,c are of different signs, so, . Hence, the roots of the equation are real.
We know that if X,Y are roots of then
Since a,c are of different signs, so, XY is negative, so, Both roots are of different signs.
Hence, answer is option D.