If the locus of a point which moves so that the line joining...

If the locus of a point which moves so that the line joining the points of contact of the tangents drawn from it to the circle $x^{2}+y^{2}=b^{2}$ touches the circle $x^{2}+$ $y^{2}=a^{2}$, is the circle $x^{2}+y^{2}=c^{2}$, then $a, b, c$ are in (A) A. P. (B) G. P. (C) H. P. (D) none of these

Key Concepts

Geometric Locus

A locus is the set of all points that satisfy a particular condition or a set of conditions. In geometry, determining the locus of a point involves finding an equation or description that represents all positions where the point can exist while fulfilling the given properties.

Tangents to a Circle

Tangents are lines that touch a circle at exactly one point. They are significant in circle geometry because, at the point of tangency, the tangent is perpendicular to the radius of the circle. This unique property is often used to establish relationships between different geometric elements and to solve problems involving circles.

Chord of Contact

The chord of contact is the line segment joining the points of tangency from an external point to a circle. This concept is useful in many geometric problems because it connects the external point with the circle’s geometry, often leading to interesting loci or envelope shapes.

Tangency Condition for a Line and a Circle

For a line to be tangent to a circle, the perpendicular distance from the center of the circle to the line must equal the radius of the circle. This condition provides an important equation that relates the line’s parameters to the circle’s parameters and is crucial in solving problems involving tangency.

Sequence Relations in Geometric Parameters

Arithmetic, geometric, and harmonic progressions are ways of relating quantities so that there is a regular pattern or ratio between them. In geometric problems, establishing that certain parameters (like radii of circles or distances) follow one of these progressions can unveil deep relationships and constraints arising from the geometric configuration.

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