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(C) Given parabola is $y^2 = 2ax$. Focus is $(a/2, 0)$ and the directrix is $x = -a/2$. Since the circle is tangent to the directrix,its radius $r$ is

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$(a/2, \pm a)$

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(C) Given parabola is $y^2 = 2ax$. Focus is $(a/2, 0)$ and the directrix is $x = -a/2$. Since the circle is tangent to the directrix,its radius $r$ is the distance from the focus $(a/2, 0)$ to the line $x = -a/2$,which is $r = |a/2 - (-a/2)| = a$. The equation of the circle with center $(a/2, 0)$ and radius $a$ is $(x - a/2)^2 + y^2 = a^2$. Substituting $y^2 = 2ax$ into the circle equation: $(x - a/2)^2 + 2ax = a^2$ $x^2 - ax + a^2/4 + 2ax = a^2$ $x^2 + ax - 3a^2/4 = 0$ Using the quadratic formula $x = \frac{-a \pm \sqrt{a^2 - 4(1)(-3a^2/4)}}{2} = \frac{-a \pm \sqrt{a^2 + 3a^2}}{2} = \frac{-a \pm 2a}{2}$. So,$x = a/2$ or $x = -3a/2$. For $x = a/2$,$y^2 = 2a(a/2) = a^2$,so $y = \pm a$. For $x = -3a/2$,$y^2 = 2a(-3a/2) = -3a^2$,which gives no real values for $y$. Thus,the points of intersection are $(a/2, \pm a)$.

Let a circle tangent to the directrix of a parabola $y^2 = 2ax$ have its center coinciding with the focus of the parabola. Then the points of intersection of the parabola and the circle are

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