Focal chord
Understanding the Focal Chord
The focal chord of a parabola is a line segment that passes through the focus of the parabola and has both endpoints on the parabola itself. To understand the focal chord in depth, we need to explore its properties, how to find its length, and its relation to the parabola's equation.
Properties of a Focal Chord
A focal chord possesses several interesting properties:
- Perpendicular to the Axis of Symmetry: Every focal chord is perpendicular to the axis of symmetry of the parabola.
- Midpoint on the Directrix: The midpoint of a focal chord lies on the directrix of the parabola.
- Length Related to the Parabola's Parameter: The length of the focal chord can be related to the parabola's parameter (the distance from the focus to the directrix).
Parabola and its Equation
Before we delve into the focal chord, let's briefly review the standard form of a parabola's equation and its elements.
For a parabola with a vertical axis of symmetry, the standard form of the equation is:
$$ y^2 = 4ax $$
Where:
- (a) is the distance from the vertex to the focus (and also from the vertex to the directrix).
- The vertex is at the origin ((0,0)).
- The focus is at ((a,0)).
- The directrix is the line (x = -a).
Finding the Length of a Focal Chord
To find the length of a focal chord, we can use the endpoints of the chord. Let's say the endpoints are ((x_1, y_1)) and ((x_2, y_2)). Since both points lie on the parabola, they satisfy the equation (y^2 = 4ax). Thus, we have:
$$ y_1^2 = 4ax_1 $$ $$ y_2^2 = 4ax_2 $$
The distance between these two points can be found using the distance formula:
$$ L = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$
Example of Finding a Focal Chord
Let's consider a parabola with the equation (y^2 = 12x). The focus of this parabola is at ((3,0)) since (a = 3). Suppose we have a focal chord with endpoints ((x_1, y_1)) and ((x_2, y_2)) where (x_1 = 4) and (x_2 = 9). We can find (y_1) and (y_2) by plugging (x_1) and (x_2) into the parabola's equation:
$$ y_1^2 = 12 \cdot 4 = 48 \Rightarrow y_1 = \pm \sqrt{48} $$ $$ y_2^2 = 12 \cdot 9 = 108 \Rightarrow y_2 = \pm \sqrt{108} $$
Choosing the positive square roots for simplicity, we get (y_1 = 4\sqrt{3}) and (y_2 = 6\sqrt{3}). Now, we can find the length of the focal chord (L):
$$ L = \sqrt{(9 - 4)^2 + (6\sqrt{3} - 4\sqrt{3})^2} $$ $$ L = \sqrt{25 + 12} $$ $$ L = \sqrt{37} $$
Table of Differences and Important Points
| Property | Description | Relevance to Focal Chord |
|---|---|---|
| Axis of Symmetry | The line that divides the parabola into two mirror-image halves | Focal chord is perpendicular to it |
| Focus | A fixed point inside the parabola used to define its shape | Focal chord passes through the focus |
| Directrix | A fixed line outside the parabola used to define its shape | Midpoint of the focal chord lies on the directrix |
| Length | The distance between the endpoints of the chord | Can be calculated using the endpoints that satisfy the parabola's equation |
Conclusion
The focal chord is a significant concept in the study of parabolas. It is directly related to the parabola's focus and directrix and can be analyzed using the parabola's equation. Understanding the properties and methods to find the length of a focal chord is essential for solving problems related to parabolas in exams.