AB is a chord of the circle x^2+y^2-4 x-2 y+1=0 subtending a right angle at the centre. Then the

step 1 So in this question AB is a chord of this circle and this cord is subtending right angle at the

AB is a chord of the circle x^2+y^2-4 x-2 y+1=0 subtending a...

$\mathrm{AB}$ is a chord of the circle $\mathrm{x}^{2}+\mathrm{y}^{2}-4 \mathrm{x}-2 \mathrm{y}+1=0$ subtending a right angle at the centre. Then the locus of the centroid of the triangle PAB, where $\mathrm{P}$ moves on the circle is (a) a parabola (b) an ellipse (c) a hyperbola (d) a circle

$\mathrm{AB}$ is a chord of the circle $\mathrm{x}^{2}+\mathrm{y}^{2}-4 \mathrm{x}-2 \mathrm{y}+1=0$ subtending a right angle at the centre. Then the locus of the centroid of the triangle PAB, where $\mathrm{P}$ moves on the circle is
(a) a parabola
(b) an ellipse
(c) a hyperbola
(d) a circle

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Key Concepts

Circle Geometry

Circle geometry uses algebraic methods to analyze circles, including determining the center and radius from a given equation. It also examines properties like chords, tangents, and angles that the chords subtend at the center, which are essential for solving many geometrical problems.

Centroid of a Triangle

The centroid of a triangle is the point where its medians intersect, calculated as the average of the triangle's vertices' coordinates. It represents the center of mass of the triangle and plays a crucial role in establishing the locus when one of the vertices moves along a defined path.

Locus of a Point

A locus is the set of points that satisfy a particular condition or series of conditions. In coordinate geometry, finding the locus involves deriving an equation that describes all possible positions of a point that meets specific geometric constraints.

Chord Subtending a Fixed Angle at the Center

A chord subtending a given angle at the center of a circle, such as a right angle, has distinctive geometric properties that relate the chord's length to the circle's radius. This relationship is significant in problems involving dynamic configurations where such chords influence the positions and properties of other points in the figure.

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