If the chords of contact of points on ${x}^{2}+{y}^{2}={a}^{2}$ with respect to the circle ${x}^{2}+{y}^{2}={b}^{2}$ touch the circle ${x}^{2}+{y}^{2}={c}^{2}$, then ${a},\ {b},\ {c}$ are in

Let a point on the circle x squared + y squared = a squared be (a cos theta, a sin theta). The equation of the chord of contact with respect to the circle x squared + y squared = b squared is x(a cos theta) + y(a sin theta) = b squared, which simplifies to x cos theta + y sin theta = b squared / a. The distance from the origin to this line must equal the radius c for it to be tangent to x squared + y squared = c squared, giving (b squared / a) = c. This means b squared = a times c, which is the condition for a, b, and c to be in a geometric progression (G.P.).

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