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(D) The focus of the parabola $y^2 = 2px$ is $S = \left(\frac{p}{2}, 0\right)$.The directrix of the parabola is $x = -\frac{p}{2}$.Since the circle is

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Both $(A)$ and $(B)$

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(D) The focus of the parabola $y^2 = 2px$ is $S = \left(\frac{p}{2}, 0\right)$.The directrix of the parabola is $x = -\frac{p}{2}$.Since the circle is centered at the focus and touches the directrix,the radius $r$ is the distance from the focus to the directrix,which is $r = \frac{p}{2} - (-\frac{p}{2}) = p$.The equation of the circle is $\left(x - \frac{p}{2}\right)^2 + y^2 = p^2$.Substitute $y^2 = 2px$ into the circle equation:$\left(x - \frac{p}{2}\right)^2 + 2px = p^2$Expanding the equation:$x^2 - px + \frac{p^2}{4} + 2px = p^2$$x^2 + px - \frac{3p^2}{4} = 0$$4x^2 + 4px - 3p^2 = 0$Factoring the quadratic equation:$(2x + 3p)(2x - p) = 0$This gives $x = \frac{p}{2}$ or $x = -\frac{3p}{2}$.For $x = \frac{p}{2}$,$y^2 = 2p\left(\frac{p}{2}\right) = p^2$,so $y = \pm p$.For $x = -\frac{3p}{2}$,$y^2 = 2p\left(-\frac{3p}{2}\right) = -3p^2$,which gives no real solution for $y$.Thus,the points of intersection are $\left(\frac{p}{2}, p\right)$ and $\left(\frac{p}{2}, -p\right)$.

Consider a circle with its centre lying on the focus of the parabola $y^2 = 2px$ such that it touches the directrix of the parabola. Then a point of intersection of the circle and the parabola is:

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