Common Tangents in Coordinate Geometry Explained

We know that if a line intersects a circle in just one point, it will touch the circle, called a tangent to the circle, and the point at which it touches the circle called as a point of contact of that tangent. Leibniz identified the tangent as the line through a pair of infinitely close points over the curve. Further, this term has been derived from the Latin word ‘tangere’ which means ‘to touch’.

The following figure shows a circle S with tangent L, and the point of contact as P:

What are Common Tangents in Coordinate Geometry?

A line which is tangent to more than one circle is defined as a common tangent. The internal common tangent and external common tangents are two types of common tangents in coordinate geometry. An internal tangent is defined as a line segment, which passes through the centre of the two circles whereas the external common tangents do not.

There are possibilities of four common tangents to two circles. The tangents can be either direct or transverse.

Assume that two circles that are dawn lie externally to each other and touch each other externally, or intersect each other in two distinct points. In these cases, exactly two direct common tangents will exist as shown in the figure below.

A common tangent is said to be transverse if the two circles lie on opposite sides of it. In the situation below, we have two circles lying externally to each other, and exactly two transverse common tangents:

For two circles touching each other externally, there will be only one transverse common tangent.                

For two intersecting circles, there is no transverse common tangent but it has two direct common tangents. Similarly, for two circles touching each other internally, exactly one direct common tangent and there is no transverse common tangent.

Important Facts

• When one circle lies inside the other completely without touching, there is no common tangent.

• When two circles touch each other internally, one common tangent can be drawn to the two circles that touch each other internally. 

• Two common tangents can be drawn to the two circles that intersects each other at two different points. 

• When two circles touch each other externally, three common tangents can be drawn to the circles.

• Four common tangents can be easily drawn to the two circles that neither touch nor intersect and one circle lies outside the other circle.

All this information along with the related conditions can be summarized as follows:

Number of Tangents                                     Condition

• 4 common tangents                           r₁+ r₂ < c₁c₂

• 3 common tangents                           r₁ + r₂ = c₁c₂

• 2 common tangents                           |r₁ - r₂| < c₁c₂ < r₁ + r₂

• 1 common tangents                           |r₁ - r₂| = c₁c₂

• no common tangents                         c₁c₂ < |r₁ - r2| 

Solved Example

1. Calculate the Equations to the Common Tangents of the Circles 

x² + y² – 2x – 4y + 4 = 0;  x² + y² + 4x – 2y + 1 = 0.

Solution: These circles lie completely outside each other. This means, there’ll be four common tangents. Let us first find the points where the direct and the transverse common tangents meet.

These points divide the line joining the centre internally and externally in the ratio of the radii.

Here C­₁ ≡ (1, 2), r₁ = 1, C₂ = (-2, 1) and r­₂ = 2.

Using the section formula helps us to get the following points:

• Point of intersection of the direct common tangents as (4,3) 

• Transverse common tangent point as (0, 5/3).

Next, we’ll find the equation of tangents from an external point.

Let any line passing through the points(4, 3) be y – 3 = m(x – 4), or mx – y – 4m + 3 = 0.

This line will touch the first and the second circle if its distance from the circle’s centre will be equal to its radius.

That is, |m(1) – 2 – 4m + 3| / \[\sqrt{m^{2}+1}\] = 1. 

This obtains the value of m = 0 and m = 3/4.

Finally, the equation of the direct common tangents (using the point-slope form) will be

y = 4 and 3x – 4y = 0.

Now, on to the transverse common tangents.

Any line through (0, 5/3) is y – 5/3 = m(x – 0) → 3mx – 3y + 5 = 0.

Doing the same thing again, we get |3m(1) – 3(2) + 5| / \[\sqrt{9m^{2}+9}\] = 1.

This gives m = -4/3 and m = ∞, giving the equation of the tangents as x = 0 and 4x + 3y = 5.