The distances of the origin, from the centres of three...

The distances of the origin, from the centres of three circles $x^{2}+y^{2}-2 \lambda x=c^{2}$ where $c$ is constant and $\lambda$ a variable parameter are in geometrical progression. Then the lengths of tangents drawn to them from any point on the circle $\mathrm{x}^{2}+\mathrm{y}^{2}=\mathrm{c}^{2}$ are in (a) A.P (b) G.P. (c) H.P. (d) not in any progression.

The distances of the origin, from the centres of three circles $x^{2}+y^{2}-2 \lambda x=c^{2}$ where $c$ is constant and $\lambda$ a variable parameter are in geometrical progression. Then the lengths of tangents drawn to them from any point on the circle $\mathrm{x}^{2}+\mathrm{y}^{2}=\mathrm{c}^{2}$ are in
(a) A.P
(b) G.P.
(c) H.P.
(d) not in any progression.

Key Concepts

Circle Equations and Center-Radius Form

A circle's equation in general form can be rewritten in center-radius form by completing the square. This conversion reveals the circle's center and radius, which are fundamental in assessing distances, tangents, and other geometric properties related to circles.

Distance Formula in Coordinate Geometry

The distance formula, derived from the Pythagorean theorem, calculates the distance between two points in the plane. It is essential for determining the distance from the origin to any point, such as the center of a circle, and forms a basis for many geometric analyses.

Tangent Length from an External Point

The length of a tangent from an external point to a circle is given by the square root of the difference between the square of the distance from the point to the circle's center and the square of the radius. This formula highlights the relationship between the external point’s location and the geometric properties of the circle.

Sequences and Progressions (A.P., G.P., H.P.)

Arithmetic, geometric, and harmonic progressions describe specific patterns in sequences based on consistent addition, multiplication, or reciprocals. Understanding these concepts is crucial when analyzing relationships where quantities follow regular progression patterns, such as distances or tangent lengths in geometric configurations.

Parametric Families of Circles

Parametric families of circles involve a variable parameter that systematically alters properties like the center and radius. Studying these variations helps in understanding how changes in a parameter can lead to different geometric outcomes and relationships among various elements, including distances and tangents.

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