Common External and Internal Tangents

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 touches both circles without crossing the segment between their 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:

This positive-length segment exists when $d > |r_2-r_1|$. At equality, the circles are internally tangent and share only the tangent line through the contact point.

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:

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

Condition: A positive-length internal tangent segment exists when $d > r_1 + r_2$. At equality, the circles are externally tangent: the two points of tangency coincide, the segment length is zero, and only one internal common tangent remains.

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:

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

Circles with Equal Radii

When two circles have equal radii ($r_1 = r_2 = r$), the common tangents simplify. The external tangent runs parallel to the line joining the centres, and the internal tangent is perpendicular to that line and passes through its midpoint.

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 the external and the internal common tangent line for the following circles. Record the distance between the centres and both radii first: the external tangent uses $l = \sqrt{d^2 - (r_2 - r_1)^2}$, the internal one uses $l = \sqrt{d^2 - (r_1 + r_2)^2}$.

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

The problems give two circles and ask for the lengths of their common tangent lines. Separate the external and internal cases before using each formula.

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

Worked Solutions

1. Calculating common tangent line lengths

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

   $$\begin{aligned} l_{\mathrm{ext}} &= \sqrt{8^2 - (3-2)^2} \\ &= \sqrt{64 - 1} \\ &= \sqrt{63} \\ &= 3\sqrt{7} \end{aligned}$$

   $$\begin{aligned} l_{\mathrm{int}} &= \sqrt{8^2 - (2+3)^2} \\ &= \sqrt{64 - 25} \\ &= \sqrt{39} \end{aligned}$$

2. Circles with different centers

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

   $$\begin{aligned} l_{\mathrm{ext}} &= \sqrt{10^2 - (4-2)^2} \\ &= \sqrt{100 - 4} \\ &= \sqrt{96} \\ &= 4\sqrt{6} \end{aligned}$$

   $$\begin{aligned} l_{\mathrm{int}} &= \sqrt{10^2 - (4+2)^2} \\ &= \sqrt{100 - 36} \\ &= \sqrt{64} \\ &= 8 \end{aligned}$$

3. Finding center distance from internal tangent length

   Given: $r_1 = r_2 = 3$, $l_{\mathrm{int}} = 8$

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

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

   $$64 = d^2 - 36$$

   $$d^2 = 100$$

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

4. Finding the value of $r$

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

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

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

   $$128 = 144 - r^2$$

   $$r^2 = 16$$

   $$r = 4$$

5. Conditions for number of common tangent lines

   Let $d$ be the distance between centers. The complete classification for two distinct, noncoincident circles is:

   Center-distance condition	Common tangents	Circle position
   $d>r_1+r_2$	$4$	externally disjoint
   $d=r_1+r_2$	$3$	externally tangent
   $\lvert r_1-r_2\rvert\lt d\lt r_1+r_2$	$2$	intersecting at two points
   $d=\lvert r_1-r_2\rvert$	$1$	internally tangent
   $d\lt \lvert r_1-r_2\rvert$	$0$	one circle strictly inside the other

   Therefore, the requested conditions are the first three rows. Coincident circles, with $d=0$ and $r_1=r_2$, are a separate case with infinitely many common tangents.