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 :

So,

So

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,

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.