Understanding the Director Circle of a Circle

The concept of the director circle is an interesting topic in the study of circles and their properties. The director circle of a given circle is a locus of points from which tangents drawn to the original circle are perpendicular to each other. This concept is particularly useful in coordinate geometry and has various applications.

Definition

The director circle of a circle with center at the origin and radius $r$ is another circle centered at the origin with radius $\sqrt{2}r$.

Equation

For a circle with the equation $x^2 + y^2 = r^2$, the equation of its director circle is:

$$ x^2 + y^2 = 2r^2 $$

Properties

The director circle has the following properties:

• It is concentric with the original circle.
• The radius of the director circle is $\sqrt{2}$ times the radius of the original circle.
• Every point on the director circle is such that the tangents from it to the original circle are perpendicular to each other.

Table of Differences

Aspect	Original Circle	Director Circle
Center	$(h, k)$	$(h, k)$
Radius	$r$	$\sqrt{2}r$
Equation	$(x-h)^2 + (y-k)^2 = r^2$	$(x-h)^2 + (y-k)^2 = 2r^2$
Tangents from a point	Not necessarily perpendicular	Perpendicular to each other

Examples

Example 1: Finding the Director Circle

Given a circle with the equation $x^2 + y^2 = 9$, find the equation of its director circle.

Solution:

The radius $r$ of the given circle is $\sqrt{9} = 3$. Therefore, the radius of the director circle is $\sqrt{2} \cdot 3 = 3\sqrt{2}$.

The equation of the director circle is:

$$ x^2 + y^2 = (3\sqrt{2})^2 $$ $$ x^2 + y^2 = 18 $$

Example 2: Tangents from a Point on the Director Circle

Consider the circle $x^2 + y^2 = 25$ and its director circle $x^2 + y^2 = 50$. Find the equations of the tangents from the point $(5, 5)$ on the director circle to the original circle.

Solution:

Since $(5, 5)$ lies on the director circle, the tangents from this point to the original circle will be perpendicular to each other. To find the equations of these tangents, we can use the point form of the tangent to a circle:

$$ (x - x_1)(x_1 - h) + (y - y_1)(y_1 - k) = r^2 $$

For the given circle, $h = 0$, $k = 0$, and $r^2 = 25$. Substituting the point $(5, 5)$, we get:

$$ (x - 5)(5) + (y - 5)(5) = 25 $$ $$ 5x - 25 + 5y - 25 = 25 $$ $$ 5x + 5y = 75 $$ $$ x + y = 15 $$

This is one tangent. Since the tangents are perpendicular, the slope of the other tangent will be the negative reciprocal of the slope of the first tangent. The slope of the first tangent is $-1$, so the slope of the second tangent is $1$. Using the point-slope form of a line, we get:

$$ y - 5 = 1(x - 5) $$ $$ y = x $$

Thus, the equations of the tangents from the point $(5, 5)$ to the circle $x^2 + y^2 = 25$ are $x + y = 15$ and $y = x$.

In conclusion, the director circle is a powerful concept that helps in understanding the geometric properties of circles and their tangents. It is particularly useful when dealing with perpendicular tangents and has applications in various fields such as physics, engineering, and computer graphics.