The tangent line of a circle can be defined as a line that intersects the circle at one point only.
Put a circle in the rectangular coordinate system.
Let be a point on a circle. The tangent line at is a line intersects the circle at only.
Let’s first find a function that represents the line.
From circle’s equation , we have
Since the line intersects the circle at only,
has only one solution.
has only one solution. i.e., its discriminant
By the definition of slope,
It follows that
Substitute (2) into (1) and solve for gives
The slope of line connecting and where is .
Since , the tangent line is perpendicular to the line connecting and .
Substitute (3) into , we have
The fact that the line intersects the circle at means
It follows that by (4),
(5) is derived under the assumption that . However, by letting in (5), we obtain two tangent lines that can not be expressed in the form of :