Mathematics · Ch 6 — Circle

Director Circle

6.3.4

Director Circle

Definition. The director circle of a given circle is the locus (path traced) of the point of intersection of a pair of mutually perpendicular tangents drawn to that circle.

Deriving its equation for $x^2+y^2=a^2$ . From Section 6.3.3, if tangents are drawn from a point $P(x_1,y_1)$ with slopes $m_1,m_2$ , then

m_1m_2=\frac{y_1^2-a^2}{x_1^2-a^2}

If the two tangents from $P$ happen to be perpendicular, then $m_1m_2=-1$ . Setting the expression above equal to $-1$ :

\frac{y_1^2-a^2}{x_1^2-a^2}=-1

y_1^2-a^2=-(x_1^2-a^2)=-x_1^2+a^2

x_1^2+y_1^2=2a^2

Since $(x_1,y_1)$ is a general point where two perpendicular tangents meet, dropping the subscripts gives the locus:

x^2+y^2=2a^2

The director circle of $x^2+y^2=a^2$ is $x^2+y^2=2a^2$ — a circle concentric with the original one, with radius $\sqrt2$ times as large. …

Figure Fig.6.10Fig. 6.10 — perpendicular tangents meeting on the director circle

What this figure shows. The circle x^2+y^2=a^2 with two mutually perpendicular tangent lines drawn from an external point, illustrating that as the point of intersection moves so as to keep the tangents perpendicular, it traces a larger concentric circle -- the director circle. …

Misc Chapter-summaryLet's Remember — chapter-end summary (8 points)

Worked out. A recap list covering: the standard, centre-radius and general forms of a circle with their centres/radii; the three cases of the general form (real circle / point circle / no real locus); the parametric form of the standard circle; the Cartesian and parametric tangent equations with the point of contact; the c^2=a^2m^2+a^2 tangency condition; the slope form of a tangent line; and the director circle x^2+y^ …

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