Let A B be a chord of the circle x^2+y^2=r^2 subtending a...

Let $A B$ be a chord of the circle $x^{2}+y^{2}=r^{2}$ subtending a right angle at the centre. Then, the locus of the centroid of the $\Delta P A B$ as $P$ moves on the circle is
(A) a parabola
(B) a circle
(C) an ellipse
(D) a pair of straight lines

Key Concepts

Circle Geometry

This concept involves the properties and relationships of circles, including definitions of chords, central angles, and the relationships between chords and the angles they subtend at the center. In problems where a chord is fixed and subtends a given angle, understanding how this chord interacts with other points on the circle is essential for determining geometric relationships.

Triangle Centroid

The centroid of a triangle, often referred to as the center of mass or barycenter, is the point where the three medians intersect. It is found by taking the average of the coordinates of the triangle's vertices. This concept is vital for analyzing how the location of the centroid changes as one or more vertices of the triangle move along a defined path, such as a circle.

Parametric Representation of Curves

A parametric representation describes the coordinates of points on a curve as functions of one or more parameters. In the context of circle geometry, a common parameterization uses trigonometric functions to express the x and y coordinates in terms of an angle. This approach is useful for translating geometric movement and relationships into algebraic forms that can be manipulated to find the locus of a point.

Locus Determination in Coordinate Geometry

Locus determination involves finding the set of all points that satisfy a given geometric condition. Using coordinate geometry, one can convert the motion or constraints of a moving point into an algebraic equation that represents a curve or figure, such as a circle, ellipse, or parabola. This technique is central to solving many problems where a point's position is defined by its relationships to other static or moving objects.

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