External Tangent Line and Internal Tangent Line

Definition of Common Tangent Lines

A common tangent line is a line that touches two circles simultaneously. There are two types of common tangent lines:

External Common Tangent Line: A line that touches both circles from the same side
Internal Common Tangent Line: A line that touches both circles from opposite sides

Concept of External Common Tangent Line

An external common tangent line is a line that touches both circles and does not intersect the line connecting the two circle centers.

External Common Tangent Line

Two external common tangent lines on two circles.

Formula for External Common Tangent Line Length

For two circles with:

First circle center: $O_1$
Second circle center: $O_2$
First circle radius: $r_1$

Length of external common tangent line:

l = \sqrt{d^2 - (r_2 - r_1)^2}

Finding the Length of External Common Tangent Line

Two circles are centered at $A(-3, 0)$ with radius $1.5$ and $B(3, 0)$ with radius $2.5$ . Find the length of the external common tangent line!

Circles with

r_1 = 1.5

and

r_2 = 2.5

Visualization of external common tangent line.

Solution:

d = \sqrt{(3-(-3))^2 + (0-0)^2} = \sqrt{36} = 6

Therefore, the length of the external common tangent line is $\sqrt{35} \text{ units}$ .

Concept of Internal Common Tangent Line

An internal common tangent line is a line that touches both circles from opposite sides and intersects the line connecting the two circle centers.

Internal Common Tangent Line

Two internal common tangent lines on two circles.

Formula for Internal Common Tangent Line Length

Length of internal common tangent line:

l = \sqrt{d^2 - (r_1 + r_2)^2}

Condition: Internal common tangent lines exist only if $d > r_1 + r_2$ (the two circles do not intersect).

Finding the Length of Internal Common Tangent Line

Two circles are centered at $P(-5, 0)$ with radius $2$ and $Q(5, 0)$ with radius $3$ . Find the length of the internal common tangent line!

Circles with

r_1 = 2

and

r_2 = 3

Visualization of internal common tangent line.

Solution:

First, check if internal common tangent lines exist:

d = \sqrt{(5-(-5))^2 + (0-0)^2} = \sqrt{100} = 10

Since the condition is satisfied, then:

l = \sqrt{d^2 - (r_1 + r_2)^2}

Therefore, the length of the internal common tangent line is $5\sqrt{3} \text{ units}$ .

Circles with Equal Radii

When two circles have equal radii ( $r_1 = r_2 = r$ ), there are special properties:

External Common Tangent Line

External Common Tangent Line for Equal Radii

External common tangent lines are parallel to the line of centers.

For $r_1 = r_2$ :

External common tangent lines are parallel to the line connecting the two centers
Length of external common tangent line is $d$ (distance between centers)

Cases of Various Circle Positions

Determine the length of external and internal common tangent lines for the following circles:

Distant Circles

The first circle is centered at $(-6, 0)$ with radius $1$ , the second circle is centered at $(6, 0)$ with radius $2$ .

Case: Distant Circles

Both types of common tangent lines exist.

Solution:

d = \sqrt{(6-(-6))^2 + (0-0)^2} = 12

Close Circles

The first circle is centered at $(-2, 0)$ with radius $1.5$ , the second circle is centered at $(2, 0)$ with radius $1.5$ .

Case: Close Circles with Equal Radii

External common tangent lines are parallel, internal ones intersect at the center.

Solution:

d = \sqrt{(2-(-2))^2 + (0-0)^2} = 4

Practice Problems

Two circles are centered at $A(-4, 0)$ with radius $2$ and $B(4, 0)$ with radius $3$ . Determine:
- Length of external common tangent line
- Length of internal common tangent line
The first circle has center with radius , the second circle has center with radius . Calculate the length of both types of common tangent lines!

Answer Key

Calculating common tangent line lengths
$d = \sqrt{(4-(-4))^2 + (0-0)^2} = 8$