The locus of the centre of a circle which passes through the...

The locus of the centre of a circle which passes through the point $(0,0)$ and cuts off a length $2 b$ from the line $x$ $=c$, is
(A) $y^{2}+2 c x=b^{2}+c^{2}$
(B) $x^{2}+c x=b^{2}+c^{2}$
(C) $y^{2}+2 c y=b^{2}+c^{2}$
(D) none of these

Key Concepts

Locus of a Point

A locus is a set of all points that satisfy one or more given conditions. In the context of circle geometry, the locus of centers refers to all the possible positions of the center that allow the circle to meet the imposed conditions such as passing through a fixed point or cutting a chord of a prescribed length on a given line.

Circle Through a Fixed Point

When a circle is required to pass through a specific point, the coordinates of that point create a relationship between the circle’s radius and its center. This condition is used to relate the distance from the center to the fixed point with the radius, forming an equation that constrains the possible locations of the circle’s center.

Chord Length and Distance Relation

In circle geometry, the length of a chord intercepted by a line is connected to the circle’s radius and the perpendicular distance from the circle’s center to the line. This relation, derived from the Pythagorean theorem, states that the chord length is twice the square root of the difference between the square of the radius and the square of this distance, thereby linking geometric quantities in an algebraic form.

Algebraic Formulation in Coordinate Geometry

Using coordinate geometry, geometric properties of the circle can be translated into algebraic equations. By combining the condition that the circle passes through a fixed point with the chord length formula, one can derive an equation representing the locus of the center of the circle. This approach facilitates the analysis and solution of the problem through algebraic manipulation.

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