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Home » Lessons » Circle » Equations of Common Tangents to Two Circles: Examples

Equations of Common Tangents to Two Circles: Examples

This lesson will cover a few examples relating to equations of common tangents to two given circles.

Example 1 Find the equation of the common tangents to the circles x2 + y2 – 2x – 4y + 4 = 0 and x2 + y2 + 4x – 2y + 1 = 0.

Solution These circles lie completely outside each other (go back here to find out why). That means, there’ll be four common tangents, as discussed previously.

We’ll first need to find the points where the direct and the transverse common tangents meet.

As discussed in the previous lesson, these points divide the line joining the centre in the ratio of the radii (internally and externally).

Let’s begin the calculations then. Here C­1 ≡ (1, 2), r1 = 1, C2 ≡ (-2, 1) and r­2 = 2.

Using the section formula, we get the point of intersection of the direct common tangents as (4, 3) and that of the transverse common tangents as (0, 5/3).

Next, we’ll use our knowledge of finding equation of tangents from an external point.

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

Now this line will touch the first (also the second) circle if its distance from the circle’s centre will be equal to its radius.

That is, |m(1) – 2 – 4m + 3|/ m2+1−−−−−−√ = 1. This gives m = 0 and m = 3/4.

We have the equation of the direct common tangents (using the point-slope form) will be y = 4 and 3x – 4y = 0.

On to the transve