If the roots of $a x^{2} + bx + c = 0$ are equal in magnitude but opposite in sign, then a possible condition is

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a < 0, c < 0

a < 0, c > 0, b > 0

a > 0, b = 0, c < 0

a > 0, b = 0, c > 0

answer is C.

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Detailed Solution

The quadratic equation provided is,

a x^{2} + bx + c = 0

Let suppose one of the roots of equation = p
Thus, other root = -p
Here,
Sum of roots

= \frac{- b}{a}

p + (- p) = \frac{- b}{a}

0 = \frac{- b}{a}

b = 0

Now, the product of roots

= \frac{c}{a}

(p) \times (- p) = \frac{c}{a}

- (p)^{2} = \frac{c}{a}

Left hand side will always be negative as the square of any value of p.
So, for this the value of a must be greater than 1 and value of c must be less than 1.
Thus, possible condition,

a > 0, b = 0, c < 0

So, option 3 is the correct option.

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