Quadratic Equations
Nature of Roots
JEE Advanced
Grade 11
Question:
Let $\alpha, \beta$ be the roots of $px^2 + qx + r = 0$, $p \neq 0$. If $p, q, r$ are in A.P. and $\frac{1}{\alpha} + \frac{1}{\beta} = 4$, then $|\alpha - \beta|$ equals:
(1) $\frac{\sqrt{34}}{9}$
(2) $\frac{2\sqrt{13}}{9}$
(3) $\frac{\sqrt{61}}{9}$
(4) $\frac{2\sqrt{17}}{9}$
Step-by-Step Solution
Key Concept: $1/\alpha + 1/\beta = (\alpha + \beta)/(\alpha\beta) = 4$ and $p, q, r$ in A.P. gives $2q = p + r$. These two conditions together fix $\alpha\beta$ and $\alpha + \beta$. Then $|\alpha - \beta| = \sqrt{(\alpha + \beta)^2 - 4\alpha\beta}$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)