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Question: If 0 \< a \< b \< c and the roots a,b of the equation ax<sup>2</sup> + bx + c = 0 are imaginary then...

If 0 < a < b < c and the roots a,b of the equation ax2 + bx + c = 0 are imaginary then –

A

| a | = | b | > 1

B

| a | = | b | < 1

C

| a | ¹ | b |

D

None of these

Answer

| a | = | b | > 1

Explanation

Solution

Roots are imaginary so

D = b2 – 4ac < 0

Roots are a = −b+b2−4ac2a\frac{- b + \sqrt{b^{2} - 4ac}}{2a}

b =−b−b2−4ac2a\frac{- b - \sqrt{b^{2} - 4ac}}{2a}

a =−b+i4ac−b22a\frac{- b + i\sqrt{4ac - b^{2}}}{2a}, b = −b−i4ac−b22a\frac{- b - i\sqrt{4ac - b^{2}}}{2a}

βˉ\bar{\beta}= a Ž | a | = |βˉ\bar{\beta}| = | b |

a | =b24a2+−b2+4ac4a2\sqrt{\frac{b^{2}}{4a^{2}} + \frac{- b^{2} + 4ac}{4a^{2}}}Žca\sqrt{\frac{c}{a}} > 1

Q c > a > 0

| a | > 1 Ž | a | = | b | > 1