it always lies in that plane
Always
Always; although that line can lie in infinitely many planes.
Unique line assumption. There is exactly one line passing through two distinct points.
If the line passing through these points is a straight line then it has a positive gradient.
Infinitely many if the 3 distinct points are collinear. Otherwise just 1.
No, a secant line does not pass through the center of a circle. A secant line is a line that intersects a circle in two distinct points. Only a line passing through the center of a circle is called a diameter.
A line that passes through a circle at two distinct points is called a secant.
Sometimes. But it could also stand for the ray or [infinite] line passing through the points A and B.
You can have an infinite number of planes passing through three collinear points.
Yes
False!
Another set of points are needed to find the slope.