an infinite number
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Yes because a line can lie in many planes so one we add one point not on that line, we define a unique plane.
They define one plane. A line is defined by two points, and it takes three points to define a plane, so two points on the line, and one more point not on the line equals one plane.
There are two possible answers; if the line is crossing the plane at an angle, then the line and the plane only intersect at one point. However, if the line is part of the plane, then the entire line intersects with the plane, and there are an infinite number of intersecting points.
In a Euclidean plane, only one.