Director Circle of a Circle – Equation and Proof

Here you will learn what is the equation of director circle of circle with proof.

Let’s begin –

Director Circle of a Circle

The equation of director circle is \(x^2\) + \(y^2\) = 2\(a^2\).

Proof : The locus of the point of intersection of two perpendicular tangents to the circle is called director circle. Let P(h, k) is the point of intersection of two tangents drawn on the circle (\(x^2\) + \(y^2\) = \(a^2\)). Then the equation of the pair of tangents is \(SS_1 = T^2\).

i.e. (\(x^2\) + \(y^2\) – \(a^2\))(\(h^2\) + \(k^2\) – \(a^2\)) = \({(hx + ky – a^2)}^2\)

As lines are perpendicular to each other then, coefficient of \(x^2\) + coefficient of \(y^2\) = 0.

\(\implies\)   [(\(h^2\) + \(k^2\) – \(a^2)-h^2\)][(\(h^2\) + \(k^2\) – \(a^2)-k^2\)] = 0

\(\implies\)  \(h^2\) + \(k^2\) = 2\(a^2\)

\(\therefore\)  locus of (h, k) is \(x^2\) + \(y^2\) = 2\(a^2\) which is the equation of the director circle

\(\therefore\)  director circle is a concentric circle whose radius is \(\sqrt{2}\) times the radius of the circle.

Note : The director circle of \(x^2 + y^2 + 2gx + 2fy + c = 0\) is

\(x^2 + y^2 + 2gx + 2fy + 2c – g^2 – f^2\) = 0

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