A parabola $y = ax^{2} + bx + c$ crosses the $x$-axis at $(\alpha,0)(\beta,0)$ both to the right of

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A parabola $y = ax^{2} + bx + c$ crosses the $x$-axis at $(\alpha,0)(\beta,0)$ both to the right of the origin. A circle also passes through these two points. The length of a tangent from the origin to the circle is :

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Sol. [Key Concept: Tangent length from $O$ $= \sqrt{power\ of\ O} = \sqrt{\alpha\beta}$; with the roots' product $\alpha\beta = \frac{c}{a}$ this is $\sqrt{c/a}$.]$$ax^{2} + bx + c = 0\frac{\alpha}{\beta}^{\alpha} \Rightarrow \alpha\beta = \frac{c}{a} $$$$\begin{matrix} (OT)^{2} & \ = (OP)(OQ) = \alpha\beta = \frac{c}{a} \ \Rightarrow \ OT & \ = \sqrt{\frac{c}{a}} \end{matrix}$$

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