Understanding Common Tangents in Geometry

A common tangent is any straight line that is simultaneously tangent to two given curves, most frequently two circles in coordinate geometry. The study of common tangents is central to the analysis of the positional relations of circles, and its treatment requires explicit geometric and algebraic calculation.

Classification of Common Tangents Between Two Circles

Let two circles have centers $C_1(x_1, y_1)$, $C_2(x_2, y_2)$ and radii $r_1 > 0$, $r_2 > 0$, respectively. The number and arrangement of common tangents depend on the value of the distance $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ relative to $r_1$ and $r_2$.

If $d > r_1 + r_2$, there exist exactly four common tangents: two external (also called direct) and two internal (also called transverse) tangents.

If $d = r_1 + r_2$, the circles touch externally, and there are exactly three common tangents: two direct and one transverse tangents, with one direct tangent acting as the common external tangent through the point of contact.

If $|r_1 - r_2| < d < r_1 + r_2$, there exist exactly two common tangents, both of which are external. The circles intersect at two points, and no transverse tangents exist.

If $d = |r_1 - r_2|,\, r_1 \neq r_2$, the circles touch internally, and there is exactly one common tangent at the point of internal contact. If $d < |r_1 - r_2|$, one circle lies entirely within the other and no common tangents exist.

Geometric Characterization of Direct and Transverse Common Tangents

A direct common tangent (sometimes called external) to two circles is a line that touches both circles such that the centers of the circles lie on the same side of the tangent. The points of contact do not lie between the centers. An internal common tangent (or transverse tangent) touches both circles, with the centers lying on opposite sides of the tangent. The tangent passes between the two circles.

Equation of the Line of Centers and Division Ratios

The centers $C_1$ and $C_2$ determine the line of centers, whose equation is given by

$\displaystyle \frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - x_1}$

or alternatively by

$\displaystyle (y - y_1)(x_2 - x_1) = (y_2 - y_1)(x - x_1).$

For direct common tangents, their point of intersection $P$ divides the segment $C_1C_2$ externally in the ratio $r_1 : r_2$. For transverse common tangents, the analogous point $Q$ divides $C_1C_2$ internally in the ratio $r_1 : r_2$.

Explicit Construction of Direct Common Tangents Between Two Circles

To derive the equations of the direct common tangents, let $P(h, k)$ divide $C_1C_2$ externally in the ratio $r_1 : r_2$. Then,

$\displaystyle h = \frac{r_1 x_2 - r_2 x_1}{r_1 - r_2}$

$\displaystyle k = \frac{r_1 y_2 - r_2 y_1}{r_1 - r_2}$

Consider a general line passing through $P(h, k)$: $y - k = m(x-h)$.

The length of the perpendicular from $C_1(x_1, y_1)$ to this line must be equal to $r_1$. The length is given by

$\displaystyle \frac{|m(x_1 - h) - (y_1 - k)|}{\sqrt{1 + m^2}} = r_1$

Squaring both sides yields

$[m(x_1 - h) - (y_1 - k)]^2 = r_1^2 (1 + m^2)$

This is a quadratic equation in $m$; solving for $m$ yields the slopes of the two direct common tangents.

The equations of the two tangents passing through $P(h, k)$ are then

$y - k = m_1(x - h)$ and $y - k = m_2(x - h)$,

where $m_1$ and $m_2$ are the roots determined above.

Explicit Construction of Transverse Common Tangents Between Two Circles

For the transverse common tangents, the point $Q$ divides the line segment $C_1C_2$ internally in the ratio $r_1 : r_2$, so that

$\displaystyle h = \frac{r_1 x_2 + r_2 x_1}{r_1 + r_2}$

$\displaystyle k = \frac{r_1 y_2 + r_2 y_1}{r_1 + r_2}$

The equation of any line passing through $Q(h, k)$ is $y - k = m(x - h)$. The length of the perpendicular from $C_1(x_1, y_1)$ to this line must again be $r_1$:

$\displaystyle \frac{|m(x_1 - h) - (y_1 - k)|}{\sqrt{1 + m^2}} = r_1$

Squaring both sides gives

$[m(x_1 - h) - (y_1 - k)]^2 = r_1^2(1 + m^2)$

Again, solving this quadratic for $m$ yields two values, corresponding to the two transverse common tangents (however, note that valid real-valued $m$ will exist only when $d > r_1 + r_2$).

Determination of Number of Common Tangents in Various Circle Configurations

If $d > r_1 + r_2$, both external and internal tangents exist: $4$ common tangents in total.

If $d = r_1 + r_2$, the circles touch externally, and only one common external tangent is possible at the point of contact: $3$ common tangents (2 direct, 1 transverse).

If $|r_1 - r_2| < d < r_1 + r_2$, the circles intersect at two points; only the two direct tangents exist. No internal (transverse) tangents are possible in this configuration.

If $d = |r_1 - r_2|,\, r_1 \neq r_2$, the circles touch internally at exactly one point and there is only one common tangent, which is internal.

Result: If $d < |r_1 - r_2|$, one circle lies inside the other and no common tangents exist.

Algebraic Condition for Common Tangency of a Line to Two Circles

Let the general equation of the common tangent be $y = mx + c$. For the tangent to be common to both circles, the length of the perpendicular from each center to the tangent must equal the respective radii. That is:

$\displaystyle \frac{|mx_1 - y_1 + c|}{\sqrt{1 + m^2}} = r_1$

$\displaystyle \frac{|mx_2 - y_2 + c|}{\sqrt{1 + m^2}} = r_2$

Considering the sign conventions and eliminating $c$, solve for $m$ to determine the slopes as per the geometric constraints for direct or transverse tangency.

Length of Common Tangent Segments

For non-intersecting and non-touching circles ($d > r_1 + r_2$), the length of each direct common tangent is given by

$\displaystyle L = \sqrt{d^2 - (r_1 - r_2)^2}$

where $d$ is the distance between centers. For the transverse (internal) common tangent, the length is

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

Invariantly, these lengths are real and positive only under the allowed geometric configurations, as discussed earlier. For an explicit worked example for pulley configurations or applications to secant/tangent segments, see specialized texts or other resources such as the Three Dimensional Geometry Overview.

Radical Axis and Its Relevance to Common Tangents

Given two circles $S_1 : x^2 + y^2 + 2g_1 x + 2f_1 y + c_1 = 0$ and $S_2 : x^2 + y^2 + 2g_2 x + 2f_2 y + c_2 = 0$, their radical axis is defined by

$S_1 - S_2 = 0$

which simplifies to the equation of a straight line in general. For intersecting circles, the radical axis passes through the points of intersection and corresponds to the common chord, whereas for non-intersecting circles, it is significant as the locus of points from which tangents drawn to both circles have equal length.

Differential Equations Overview also involves locus-based arguments that share structural similarities with radical axis logic in geometry.

Example: Direct Common Tangent Computation

Given: Circle $C_1$ at $(0, 0)$, radius $3$; Circle $C_2$ at $(8, 0)$, radius $1$.

The point $P(h, k)$ divides $C_1C_2$ externally in the ratio $r_1 : r_2 = 3:1$.

$h = \frac{3 \cdot 8 - 1 \cdot 0}{3 - 1} = \frac{24}{2} = 12$

$k = \frac{3 \cdot 0 - 1 \cdot 0}{3 - 1} = 0$

The general equation of the tangent through $(12, 0)$ is $y = m(x - 12)$.

The perpendicular from $(0, 0)$ to this line is

$\displaystyle \frac{|m \cdot 0 - 0 + 12m|}{\sqrt{1 + m^2}} = 3$

$\displaystyle \frac{|12m|}{\sqrt{1 + m^2}} = 3$

$|12m| = 3\sqrt{1 + m^2}$

$144m^2 = 9(1 + m^2)$

$144m^2 = 9 + 9m^2$

$144m^2 - 9m^2 = 9$

$135m^2 = 9$

$m^2 = \frac{9}{135} = \frac{1}{15}$

Therefore $m = \pm\frac{1}{\sqrt{15}}$

Result: The equations of the two direct common tangents are $y = \frac{1}{\sqrt{15}}(x-12),\quad y = -\frac{1}{\sqrt{15}}(x-12)$.

For further study of loci and relationships involving tangents and secants, refer to the Sets, Relations And Functions resource.