Normal
Understanding the Concept of "Normal" in the Context of Circles
The term "normal" in mathematics often refers to a line or vector that is perpendicular to a surface or another line. In the context of circles, a normal is a line that is perpendicular to the tangent of the circle at the point of contact. The normal passes through the center of the circle because the radius drawn to the point of contact with the tangent is perpendicular to the tangent.
Properties of a Normal to a Circle
- It is perpendicular to the tangent at the point of contact.
- It passes through the center of the circle.
- It bisects the chord of contact of tangents from an external point.
Equation of the Normal
Given a circle with the equation $(x - h)^2 + (y - k)^2 = r^2$ and a point $P(x_1, y_1)$ on the circle, the equation of the normal at point $P$ can be derived using the slope of the radius and the negative reciprocal of the slope of the tangent.
The slope of the radius (which is the same as the slope of the normal) is given by:
$$ m_{\text{normal}} = \frac{y_1 - k}{x_1 - h} $$
Since the normal is a straight line, its equation can be written in point-slope form:
$$ y - y_1 = m_{\text{normal}}(x - x_1) $$
Substituting the slope of the normal, we get:
$$ y - y_1 = \frac{y_1 - k}{x_1 - h}(x - x_1) $$
Differences and Important Points
Here is a table summarizing the differences between a tangent and a normal to a circle:
| Property | Tangent | Normal |
|---|---|---|
| Definition | A line that touches the circle at exactly one point. | A line perpendicular to the tangent at the point of contact. |
| Slope | The slope of the tangent is the negative reciprocal of the slope of the radius at the point of contact. | The slope of the normal is the same as the slope of the radius at the point of contact. |
| Relation to Circle | Touches the circle without intersecting it (except at the point of tangency). | Passes through the center of the circle. |
| Equation Form | $y - y_1 = m_{\text{tangent}}(x - x_1)$ | $y - y_1 = m_{\text{normal}}(x - x_1)$ |
Examples
Example 1: Finding the Equation of the Normal
Given a circle with the equation $(x - 3)^2 + (y + 2)^2 = 25$ and a point of contact $P(6, -1)$, find the equation of the normal at $P$.
Solution:
The center of the circle is $(h, k) = (3, -2)$ and the radius is $r = 5$.
The slope of the normal is the same as the slope of the radius:
$$ m_{\text{normal}} = \frac{-1 - (-2)}{6 - 3} = \frac{1}{3} $$
Using the point-slope form of the equation of a line:
$$ y - (-1) = \frac{1}{3}(x - 6) $$
Simplifying:
$$ y + 1 = \frac{1}{3}x - 2 $$
$$ y = \frac{1}{3}x - 3 $$
Therefore, the equation of the normal at point $P(6, -1)$ is $y = \frac{1}{3}x - 3$.
Example 2: Normal Bisecting a Chord of Contact
Given a circle with the equation $x^2 + y^2 = 16$ and an external point $A(8, 6)$, show that the normal at the point of contact bisects the chord of contact of tangents from $A$.
Solution:
Let the points of contact of the tangents from $A$ to the circle be $B$ and $C$. The line segment $BC$ is the chord of contact. The normal at $B$ and $C$ will pass through the center of the circle, which is the origin $(0, 0)$ in this case.
Since the normal passes through the center and the point of contact, it will bisect the chord of contact by symmetry, as the center of the circle is equidistant from all points on the circle.
Thus, the normal at any point of contact will bisect the chord of contact formed by tangents from an external point.
In summary, the concept of "normal" in the context of circles is closely related to the geometric properties of the circle, such as tangents, radii, and chords. Understanding the properties and equations of normals is essential for solving problems related to circle geometry.