If a line and a circle are placed on a plane, there are three possibilities in terms of the number of points of intersection between them. First, the line intersects the circle at two distinct points. (See Figure 1 below.)

**Figure1**

Second, the line touches the circle, that is, the line intersects the circle at one and only one point. (Figure 2). The point where the line touches the circle is called * the point of tangency*. The line that touches the circle is called a

*.*

**tangent line****Figure 2**

Third, the line does not intersect the circle. (Figure 3)

**Figure 3**

Suppose that the equation of the circle is given, namely x^{2} + y^{2} = R^{2}. [This circle is centered at the origin of the Cartesian plane and has radius R.] **What are the equations of tangent lines to the circle?**

**Case 1: The slope of the tangent line is given**

If the slope of the tangent line is *m*, then the equation of the tangent line is:

……………………………………………………………………………………………………………………………………………………………….. (1)

**Example 1**

Given the circle having the equation x^{2} + y^{2} = 9, find the equation of the tangent line whose slope is 2. Also, find the coordinates of the points of tangency!

**Answer**

The equation of the circle can be rewritten as x^{2} + y^{2} = 3^{2}. From this, we know the radius of the circle, which is R = 3. To find the equation of the tangent line whose slope is 2, substitute m = 2 and R = 3 into (1). Consequently,

There are two lines that touch the circle, namely g_{1} and g_{2}, which are described by the equations:

g_{1} :

g_{2} :

**Figure 4**

In Figure 4, P is the point where the line g_{1} touches the circle and Q is the point where the line g_{2} touches the circle. To determine the coordinates of P, substitute y in the equation of g_{1}, i.e. y = 2x + 3√5, into the equation of the circle x^{2} + y^{2} = 9. It follows that:

This quadratic equation has a double root . To obtain the ordinate of P, replace x in the equation of g_{1} with . As a consequence:

Thus,

To determine the coordinates of Q, substitute y in the equation of g_{2}, i.e. y = 2x – 3√5 into the equation of the circle x^{2} + y^{2} = 9. It follows that:

This quadratic equation has a double root . To obtain the ordinate of Q, replace x in the equation of g_{2} with . As a consequence:

Therefore,

**Below is the equation of the tangent line to a circle centered at (α,β) if the slope of the tangent line is m:**

……………………………………………………………………………………………………………………………………………. (2)

**Example 2**

Let L be a circle centered at (-4,1) with radius 3. The lines g_{1} and g_{2} touch L and have a slope of 2. Find the equations of the tangent lines.

**Answer**

In this case, α = -4, β = 1, R = 3, and m = 2. Substituting the values into (2), we get:

Thus, the equations of g_{1} and g_{2} are:

g_{1} :

g_{2} :

**Figure 5**

**Case 2: The coordinates of the point of tangency is given**

This case is elaborated in the article The Equations of The Tangent Lines to A Circle (2).