What are Tangents?
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.
FAQs on Common Tangents in Cordinate Geometry
1. What is a common tangent in coordinate geometry?
A common tangent is a straight line that touches two different curves (typically circles or parabolas) at distinct points. In the context of coordinate geometry, we study the conditions under which these tangents exist and how to find their equations. The line does not cross into the interior of either curve at the point of contact, known as the point of tangency.
2. What are the different types of common tangents to two circles?
There are generally two types of common tangents that can be drawn to two non-intersecting circles:
- Direct Common Tangent (DCT): This is a tangent where both circles lie on the same side of the line. The points of tangency are on the same side relative to the line connecting the centers of the circles.
- Transverse Common Tangent (TCT): This is a tangent where the two circles lie on opposite sides of the line. The tangent line intersects the line segment connecting the centers of the two circles.
3. How does the position of two circles determine the number of possible common tangents?
The number of common tangents depends on the distance between the centers of the two circles (d) compared to their radii (r₁ and r₂):
- When the circles are separate (d > r₁ + r₂): Four common tangents are possible (2 direct and 2 transverse).
- When the circles touch externally (d = r₁ + r₂): Three common tangents are possible (2 direct and 1 transverse).
- When the circles intersect at two points (r₁ - r₂ < d < r₁ + r₂): Two common tangents are possible (both are direct).
- When the circles touch internally (d = r₁ - r₂): Only one common tangent is possible.
- When one circle is completely inside another (d < r₁ - r₂): No common tangents can be drawn.
4. What is the general method to find the equation of a common tangent?
To find the equation of a common tangent, you can assume the equation of the tangent line is y = mx + c. The core principle is that the perpendicular distance from the center of a circle to a tangent line must be equal to its radius. You apply this condition for both circles, which gives you two equations in terms of 'm' and 'c'. Solving these simultaneous equations will provide the values for the slope (m) and the y-intercept (c), thus defining the equation of the common tangent(s).
5. How does finding the equation for a transverse common tangent differ from a direct one?
The primary difference lies in the geometric setup used to find the equation. For a direct common tangent, the centers of both circles lie on the same side of the tangent line. For a transverse common tangent, the centers lie on opposite sides. This distinction is crucial when setting up the distance formula conditions. Another method involves finding the point of intersection of the tangents, which for TCTs divides the line joining the centers internally and for DCTs divides it externally.
6. Why is calculating the length of a common tangent important?
Calculating the length of a common tangent (the distance between the two points of tangency) is important in various geometric and physics problems, such as calculating the length of a belt wrapped around two pulleys. It provides a precise measurement for constructing systems involving tangential connections. The length of a direct common tangent is given by the formula √[d² - (r₁ - r₂)²], and for a transverse common tangent, it is √[d² - (r₁ + r₂)²], where 'd' is the distance between the centers.
7. What are some key properties of a tangent relevant to this topic?
Understanding these properties is fundamental to solving problems on common tangents:
- A tangent touches a circle at exactly one point.
- The radius of the circle drawn to the point of tangency is always perpendicular (at a 90° angle) to the tangent line.
- The lengths of two tangents drawn from the same external point to a circle are equal. This property is used in deriving formulas for the lengths of common tangents.
