Any straight line which meets a particular geometrical figure at a point, just touching the figure without intersecting it, is called the tangent to the geometrical figure at that point.

For example, suppose we have a circle as shown in the image below. The line AB is a tangent because it touches but does not intersect the circle, while the line CD is not a tangent since it intersects the circle at point C.
Tangent to a circle
The following are some more geometrical figures with tangents to them at some point.

Tangent to a parabola
Tangent to an ellipse

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