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.

That means

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

or

.

Hence,

.

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 :