Question

Which of the following cannot be true for the given quadratic equation $a{x}^2+bx+c=0;c\ne 0$ if ratio of roots of this equation is equal to it's reciprocal?

• A. $b\ne 0$ and $c$ is positive when $a>0$.
• B. $b=0$ and $c$ is negative when $a>0$.
• C. $b\ne 0$ and $c$ is negative when $a<0$.
• D. $b=0$ and $c$ is negative when $a<0$.

Answer 1

The correct option is D $b=0$ and $c$ is negative when $a<0$.

Let $p,q$ be the roots of the given equation.
According to the given condition,
$\frac{p}{q}=\frac{q}{p}\Rightarrow p^2=q^2\Rightarrow p=\pm q$

Case 1:
$a>0$ and $p=q$, $b\ne 0$ and $c$ is positive.

Case 2:
$a>0$ and $p=-q$, $b=0$ and $c$ is negative.

Case 3:
$a<0$ and $p=q$, $b\ne 0$ and $c$ is negative.

Case 4:
$a<0$ and $p=-q$, $b=0$ and $c$ is positive.