Tangent
Understanding the Concept of a Tangent
A tangent to a circle is a straight line that touches the circle at exactly one point. This point is known as the point of tangency. Tangents are of significant interest in geometry because they have a unique relationship with circles and other curves.
Properties of a Tangent
Here are some key properties of a tangent to a circle:
- A tangent to a circle is perpendicular to the radius at the point of tangency.
- A tangent never crosses the circle, it just touches it.
- There can be an infinite number of tangents to a circle since there are infinite points on the circle's circumference.
Tangent Formulas
The equation of the tangent to a circle can be found using the slope of the radius or the general equation of the circle. If the circle's equation is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius, and the point of tangency is $(x_1, y_1)$, then the equation of the tangent line can be written as:
$$(x - h)(x_1 - h) + (y - k)(y_1 - k) = r^2$$
Alternatively, if the slope of the radius to the point of tangency is $m$, then the slope of the tangent will be $-1/m$ (since the tangent is perpendicular to the radius), and the equation of the tangent line can be written as:
$$y - y_1 = -\frac{1}{m}(x - x_1)$$
Table of Differences and Important Points
| Aspect | Radius | Tangent |
|---|---|---|
| Definition | A line segment from the center of the circle to any point on the circle. | A line that touches the circle at exactly one point. |
| Number of Points of Contact | One (the endpoint on the circle's circumference). | One (the point of tangency). |
| Relationship with Circle | Always perpendicular to the tangent at the point of contact. | Always perpendicular to the radius at the point of contact. |
| Length | Fixed for a given circle (equal to the radius of the circle). | Not fixed; can extend infinitely in both directions. |
Examples
Example 1: Finding the Equation of a Tangent
Given a circle with the equation $(x - 3)^2 + (y + 2)^2 = 25$ and a point of tangency at $(4, -2)$, find the equation of the tangent.
Solution:
First, we find the slope of the radius by connecting the center $(3, -2)$ to the point of tangency $(4, -2)$:
$$m = \frac{-2 - (-2)}{4 - 3} = \frac{0}{1} = 0$$
Since the slope of the radius is $0$, the slope of the tangent will be undefined (perpendicular to a horizontal line). Therefore, the tangent is a vertical line passing through $(4, -2)$, and its equation is:
$$x = 4$$
Example 2: Using the Perpendicularity Property
Given a circle with center at $(0, 0)$ and radius $5$, and a tangent that touches the circle at $(3, 4)$, verify that the tangent is perpendicular to the radius.
Solution:
The slope of the radius connecting the center to the point $(3, 4)$ is:
$$m_{radius} = \frac{4 - 0}{3 - 0} = \frac{4}{3}$$
The slope of the tangent line is the negative reciprocal of the radius' slope:
$$m_{tangent} = -\frac{1}{m_{radius}} = -\frac{3}{4}$$
Multiplying the slopes of the radius and the tangent:
$$m_{radius} \times m_{tangent} = \frac{4}{3} \times -\frac{3}{4} = -1$$
Since the product of the slopes is $-1$, this confirms that the radius and the tangent are perpendicular to each other.
Understanding tangents is crucial for solving problems related to circles, especially in geometry and trigonometry. Remembering the properties and formulas associated with tangents can greatly aid in solving such problems efficiently.