Director circle
Director Circle
The concept of the director circle is associated with conic sections, such as ellipses, hyperbolas, and parabolas. In the context of hyperbolas, the director circle can be defined as the locus of points from which tangents drawn to the hyperbola are perpendicular to each other.
Definition
For a given hyperbola, the director circle is a circle centered at the center of the hyperbola with a radius equal to the distance between the center and the vertices of the hyperbola.
Equation of Director Circle for Hyperbola
Consider the standard equation of a hyperbola centered at the origin:
$$ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $$
For this hyperbola, the equation of the director circle is:
$$ x^2 + y^2 = a^2 $$
Here, $a$ is the distance from the center to a vertex along the transverse axis.
Properties of Director Circle
- The director circle is used to find the pair of tangents to the hyperbola that are perpendicular to each other.
- The radius of the director circle is equal to the semi-major axis of the hyperbola.
- The director circle passes through the vertices of the hyperbola.
Differences and Important Points
| Property | Hyperbola | Director Circle |
|---|---|---|
| Definition | A set of all points where the difference of distances to two fixed points (foci) is constant. | The locus of points from which tangents to the hyperbola are perpendicular. |
| Equation | $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ | $x^2 + y^2 = a^2$ |
| Center | At the origin (0,0) | At the origin (0,0) |
| Radius | Not applicable | Equal to the semi-major axis $a$ of the hyperbola |
| Tangents | Can have different slopes | Always perpendicular to each other |
Examples
Example 1: Finding the Director Circle
Given the hyperbola $\frac{x^2}{16} - \frac{y^2}{9} = 1$, find the equation of the director circle.
Solution:
The given hyperbola is in the standard form, where $a^2 = 16$ and $b^2 = 9$. Therefore, $a = 4$.
The equation of the director circle is:
$$ x^2 + y^2 = a^2 $$
Substituting $a = 4$, we get:
$$ x^2 + y^2 = 16 $$
Example 2: Tangents from a Point on the Director Circle
Given the hyperbola $\frac{x^2}{25} - \frac{y^2}{16} = 1$ and a point P(3, 4) on the director circle, find the equations of the tangents from P to the hyperbola.
Solution:
The director circle for the given hyperbola is $x^2 + y^2 = 25$. Since P(3, 4) lies on the director circle, the tangents from P to the hyperbola will be perpendicular to each other.
To find the equations of the tangents, we use the tangent formula for a hyperbola:
$$ \frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1 $$
For the given hyperbola, $a^2 = 25$ and $b^2 = 16$. The coordinates of point P are $(x_1, y_1) = (3, 4)$.
Substituting these values, we get two equations representing the tangents:
$$ \frac{3x}{25} - \frac{4y}{16} = 1 \quad \text{and} \quad \frac{3x}{25} + \frac{4y}{16} = 1 $$
Simplifying, we get the equations of the tangents:
$$ \frac{3x}{25} - \frac{y}{4} = 1 \quad \text{and} \quad \frac{3x}{25} + \frac{y}{4} = 1 $$
These are the equations of the tangents from point P to the hyperbola that are perpendicular to each other.
In conclusion, the director circle is a powerful tool in the study of hyperbolas, particularly when dealing with tangents and normal lines. Understanding its properties and equations is essential for solving problems related to hyperbolas in exams and practical applications.