If are the roots of the equation and , then equals

JEE MainMathematicsEasyQuadratic Equations

If $cot\theta,cosec\theta$ are the roots of the equation
$ax^{2} + bx + c = 0$ and $\Delta = b^{2} - 4ac$ , then $a^{4}$ equals

Show answer and solution

cosec\theta + cot\theta = \frac{- b}{a}

cosec\theta \cdot cot\theta = \frac{c}{a}

{cosec}^{2}\theta - \cot^{2}\theta = 1

, we get

cosec\theta - cot\theta = - \frac{a}{b}

\therefore cosec\theta = \frac{- 1}{2}(\frac{b}{a} + \frac{a}{b})

cot\theta = \frac{1}{2}(\frac{a}{b} - \frac{b}{a})

Thus,

- \frac{1}{4}(\frac{a^{2}}{b^{2}} - \frac{b^{2}}{a^{2}}) = \frac{c}{a}

\Rightarrow a^{4} - b^{4} = - 4acb^{2}

\Rightarrow a^{4} = b^{2}(b^{2} - 4ac) = b^{2}\Delta

Source: Grass-G / M-16 Mathematics Paper 25 Nov First.docx

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