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:

1. External Common Tangent Line: A line that touches both circles from the same side
2. 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.

Formula for External Common Tangent Line Length

For two circles with:

• First circle center: $O_1$
• Second circle center: $O_2$
• First circle radius: $r_1$
• Second circle radius: $r_2$
• Distance between centers: $d$

Length of external common tangent line:

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!

Solution:

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.

Formula for Internal Common Tangent Line Length

Length of internal common tangent line:

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!

Solution:

First, check if internal common tangent lines exist:

Since the condition is satisfied, then:

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

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 $2$.

Solution:

Close Circles

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

Solution:

Practice Problems

1. 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

2. The first circle has center $(0, 0)$ with radius $4$, the second circle has center $(10, 0)$ with radius $2$. Calculate the length of both types of common tangent lines!

3. Two identical circles each have radius $3 \text{ cm}$. If the length of the internal common tangent line is $8 \text{ cm}$, determine the distance between the two circle centers!

4. Circle $A$ is centered at $(-5, 0)$ with radius $r$, and circle $B$ is centered at $(7, 0)$ with radius $2r$. If the length of the external common tangent line is $4\sqrt{8}$, determine the value of $r$!

5. Determine the conditions for two circles to have:

   • Exactly two common tangent lines
   • Exactly three common tangent lines
   • Exactly four common tangent lines

Answer Key

1. Calculating common tangent line lengths

   $$d = \sqrt{(4-(-4))^2 + (0-0)^2} = 8$$

   $$l_{external} = \sqrt{8^2 - (3-2)^2} = \sqrt{64 - 1} = \sqrt{63} = 3\sqrt{7}$$

   $$l_{internal} = \sqrt{8^2 - (2+3)^2} = \sqrt{64 - 25} = \sqrt{39}$$

2. Circles with different centers

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

   $$l_{external} = \sqrt{10^2 - (4-2)^2} = \sqrt{100 - 4} = \sqrt{96} = 4\sqrt{6}$$

   $$l_{internal} = \sqrt{10^2 - (4+2)^2} = \sqrt{100 - 36} = \sqrt{64} = 8$$

3. Finding center distance from internal tangent length

   Given: $r_1 = r_2 = 3$, $l_{internal} = 8$

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

   $$8 = \sqrt{d^2 - 6^2}$$

   $$64 = d^2 - 36$$

   $$d^2 = 100$$

   $$d = 10 \text{ cm}$$

4. Finding the value of $r$

   Given: $d = 12$, $r_1 = r$, $r_2 = 2r$, $l_{external} = 4\sqrt{8}$

   $$4\sqrt{8} = \sqrt{12^2 - (2r - r)^2}$$

   $$16 \cdot 8 = 144 - r^2$$

   $$128 = 144 - r^2$$

   $$r^2 = 16$$

   $$r = 4$$

5. Conditions for number of common tangent lines

   • Exactly $2$ tangent lines: The two circles intersect at two points
   • Exactly $3$ tangent lines: The two circles are tangent (internally or externally)
   • Exactly $4$ tangent lines: The two circles are separate (do not intersect)