Let $p$ and $q$ be real numbers such that the parabola $y = x^{2} - 2px + q$ has no common point wit

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Let $p$ and $q$ be real numbers such that the parabola $y = x^{2} - 2px + q$ has no common point with the x -axis. Let there exist points A and B on the parabola such that AB is parallel to the x -axis and $\angle AOB = 90^{\circ}$ (' O ' is origin), then possible values of q is/ are

Accepted Answer

$D < 0 \Rightarrow p^{2} - q < 0$$$\begin{matrix} \frac{\lambda\lambda}{x_{1}x_{2}} & \ = - 1 \Rightarrow \ x_{1}x_{2} = - \lambda^{2} \ \lambda & \ = x^{2} - 2px + q \Rightarrow x^{2} - 2px + q - \lambda = 0x_{x_{2}}^{x_{1}} \ \Rightarrow \ q - \lambda & \ = - \lambda^{2}\ \Rightarrow \ q = \lambda - \lambda^{2} \leq \frac{1}{4} \end{matrix}$$

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