Can the foci and the vertices of a hyperbola be the same points?  It seems to me the answer is YES.
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A parabola has the property that any point on the curve is the same distance from the focus as it is from the directrix. If the focus were at the vertex then the directrix would be tangential to the vertex and all points would be zero distance from the directrix and focus, so the parabola collapses to a single point, and could not be classed as a parabola. So, no, the focus cannot coincide with the vertex!

 

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