Director Circle: Formula and Equation

The director circle is a fundamental concept in coordinate geometry, particularly in the study of conic sections such as circles and ellipses. It is a specialized circle associated with a given conic section and plays a significant role in various geometric constructions and properties.

Director Circle

The director circle is specifically associated with an ellipse and is crucial in understanding certain geometric properties related to ellipses. It is defined as the circle on which all the pairs of tangents to the ellipse are of equal length.

The locus of the point through which perpendicular tangents are drawn to a given circle S = 0 is a circle called the director circle of the circle S = 0.

If two tangents are drawn from any point on the Director circle to the circle, then the angle between the tangents is 90^o.

The equation of director circle of the circle (S) : $x^2+y^2=a^2$ is $\mathrm{x}^2+\mathrm{y}^2=2 \mathrm{a}^2$

Proof:
The equation of any tangent to the circle $x^2+y^2=a^2$ is

$
y=m x+a \sqrt{\left(1+m^2\right)} \quad[\text { slope form }] \quad \ldots \text { (i) }
$
Point $\mathrm{P}(\mathrm{h}, \mathrm{k})$ is point of intersection of two tangents, then point P lies on the Eq (i)

$
\begin{gathered}
k=m h+a \sqrt{\left(1+m^2\right)} \\
\text { or } \quad(k-m h)^2=a^2\left(1+m^2\right) \\
\text { or } \quad m^2\left(h^2-a^2\right)-2 m k h+k^2-a^2=0
\end{gathered}
$
This is quadratic equation in $m$, let two roots are $m_1$ and $m_2$
But tangents are perpendiculars, therefore $m_1 m_2=-1$

$
\Rightarrow \frac{k^2-a^2}{h^2-a^2}=-1 \Rightarrow k^2-a^2=-h^2+a^2 \Rightarrow h^2+k^2=2 a^2
$
Hence, locus of $P(h, k)$ is $x^2+y^2=2 a^2$

So the director circle for any circle is a circle which is concentric with the given circle and whose radius is $\sqrt{2}$ times the radius of the given circle. This fact is applicable for the circles that have non-origin centres as well.

Solved Examples Based on Director Circle

Example 1: What is the equation of director circle of the circle $x^2+y^2+2 x-4 y+1=0$
1) $x^2+y^2+2 x-4 y+2=0$
2) $x^2+y^2+2 x-4 y+3=0$
3) $x^2+y^2+2 x-4 y-2=0$
4) $x^2+y^2+2 x-4 y-3=0$

Solution

$
x^2+y^2+2 x-4 y+1=0
$

Centre : $(-1,2)$, Radius $=2$
As we know, director circle has same centre and $\sqrt{2}$ times the radius of original circle
Hence, the equation of director circle is

$
\begin{aligned}
& (x+1)^2+(y-2)^2=(2 \sqrt{2})^2 \\
& x^2+y^2+2 x-4 y-3=0
\end{aligned}
$

Hence, the answer is the option 4.

Example 2: What is the radius of the director circle of $x^2+y^2-6 x+6 y+11=0$ ?
1) $\sqrt{7}$
2) $\sqrt{14}$
3) 7
4) 14

Solution

As we have learned
The radius of the Director Circle is $\sqrt{2}$ times the radius of the given circle.
Now,
The radius for the given circle $=$

$
\sqrt{3^2+3^2-11}=\sqrt{18-11}=\sqrt{7}
$
Thus, the radius of the Director Circle $=\sqrt{7} \times \sqrt{2}=\sqrt{14}$.
Hence, the answer is the option 2 .

Example 3: A director circle is drawn to the circle $\mathrm{x}^2+\mathrm{y}^2=4$. Then a director circle is drawn to this director circle. If this process is performed n a number of times the equation of the last of these circles is $\mathrm{x}^2+\mathrm{y}^2=64 \cdot$ Then n is equal to:

1) 7
2) 8
3) 5
4) 4

Solution
The radius of the $\mathrm{n}^{\text {th }}$ circle $=4.2^{\mathrm{n}}=64$

$
\Rightarrow \mathrm{n}=4
$

Hence, the answer is the option (4).

Example 4: The locus of the midpoint of the line segment joining the focus to a moving point on the parabola $\mathrm{x}^2=4$ ay is another parabola with director circle.

1) $y=-a$
2) $\mathrm{y}=\frac{-\mathrm{a}}{2}$
3) $y=0$
4) $\mathrm{y}=\frac{\mathrm{a}}{2}$

Solution
Focus of the parabola $x^2=4$ ay $\ldots$ (1) is $(0$, a)
Let $\mathrm{P}(\alpha, \beta)$ be the midpoint of the line segment joining the focus to the variable point $(\mathrm{x}, \mathrm{y})$ on the parabola.
Then $\alpha=\frac{\mathrm{x}}{2}, \beta=\frac{\mathrm{y}+\mathrm{a}}{2} \Rightarrow \mathrm{x}=2 \alpha, \mathrm{y}=2 \beta-\mathrm{a}$
$\therefore \operatorname{From}(1), 4 \alpha^2=4 a(2 \beta-a)$
Required locus is $\mathrm{x}^2=\mathrm{a}(2 \mathrm{y}-\mathrm{a})=2 \mathrm{a}\left(\mathrm{y}-\frac{\mathrm{a}}{2}\right)$
Shift the origin to $\left(0, \frac{a}{2}\right), \mathrm{x}=\mathrm{X}, \mathrm{y}=\mathrm{Y}+\frac{\mathrm{a}}{2}$
Then the locus is $\mathrm{X}^2=2 \mathrm{aY}$
This represents parabola whose directrix is $\mathrm{Y}=-\frac{\mathrm{a}}{2}$ or $\mathrm{y}=0$
Hence, the answer is the option (3).

Example 5: The angle between the tangents drawn from a point on the circle $x^2+y^2=50$ to the circle $x^2+y^2=25$ is

1) $45^{\circ}$
2) $60^{\circ}$
3) $90^{\circ}$
4) $120^{\circ}$

Solution
Clearly, $x^2+y^2=50$ is the director circle of the circle
$x^2+y^2=25$ So, the angle between the tangents is a right angle.
Hence, the answer is the option (3).

Summary

The director circle is a vital concept in coordinate geometry, particularly in the study of ellipses. It represents a circle associated with a given ellipse and provides insights into the geometric properties and relationships involving tangents. By understanding and applying the director circle, one can simplify complex geometric problems and gain deeper insights into the nature of ellipses and their tangents.