Problem 70 - A Complete Resource Book in Mathematics for JEE Main

Question

If a chord \(A B\) subtends a right angle at the centre of a given circle, then the locus of the centroid of the triangle \(P A B\) as \(P\) moves on the circle is a/an (A) parabola (B) ellipse (C) hyperbola (D) circle

Step-by-Step Solution

Answer

The locus is a circle.

Step 1: Understanding the problem

We need to find the locus of the centroid of triangle \(PAB\) where \(P\) moves on a circle, and chord \(AB\) subtends a right angle at the center of that circle. Let the center of the circle be \(O\), and radius \(r\).

Step 2: Properties of the chord

Since chord \(AB\) subtends a right angle at the center \(O\), we know that \(OA = OB\) are radii of the circle, and \(\angle AOB = 90^\circ\). The chord \(AB\) splits the circle into two semicircles.

Step 3: Centroid of Triangle PAB

The centroid \(G\) of the triangle \(PAB\) is given by the formula \((x_G, y_G) = \left(\frac{x_P + x_A + x_B}{3}, \frac{y_P + y_A + y_B}{3}\right)\). As \(P\) moves on the circle, \(G\) traces a path based on the positions of \(A\), \(B\), and \(P\).

Step 4: Analyze locus as P moves around circle

The key is to consider the behavior as \(P\) traces its path. Since \(P\) is constantly on the circle, if \(AB\) always subtends a right angle, triangle \(PAB\) becomes an isosceles right triangle when viewed from certain positions, further indicating symmetry. The locus of centroid \(G\) behaves as a circle when all such triangles are superposed.

Step 5: Derive the locus shape

To determine the accurate curve traced by \(G\), note that whenever \(P\) is on the circle opposite to the arc subtended by \(AB\), \(G\) moves through positions such that its path symmetrical and equidistant from the center. This constant distance and symmetrical path suggest that \(G\) does not depend linearly; rather, it describes another simpler circle inside the primary one.

Key Concepts

Chord Properties

Understanding the properties of chords in a circle is crucial to solving complex geometry problems. A chord in a circle is simply a line segment whose endpoints lie on the circle. In this exercise, chord \(AB\) is special because it subtends a right angle at the center \(O\) of the circle. This means that triangle \(AOB\), formed by the radii \(OA\) and \(OB\) and the chord \(AB\), is a right triangle with \(\angle AOB = 90^\circ\).
Here are some key properties to note:

• Since \(\angle AOB\) is \(90^\circ\), it implies by the angle subtended theorem that \(AB\) is the diameter of the semicircle opposite to the angle.
• The radius of the circle \(r\) plays a crucial role in defining the chord's properties, particularly its length.
• A right angle subtended by a chord is significant because it creates symmetry around the circle, influencing other geometric properties like locus forms.

These properties allow us to understand how the placement and movement of point \(P\) on the circle affect the locus of any geometric center, such as the centroid discussed in this exercise.

Centroid of a Triangle

The centroid is a point inside a triangle where the three medians intersect. It is often considered the triangle's center of mass. In triangle geometry, the centroid is denoted as \(G\), and its coordinates \(x_G, y_G\) are determined by the average of the vertices’ coordinates:

\[(x_G, y_G) = \left(\frac{x_P + x_A + x_B}{3}, \frac{y_P + y_A + y_B}{3}\right)\]

This means the centroid lies at one-third the distance from each vertex along the median. In this exercise, as point \(P\) moves along the circle, \(A\) and \(B\) remain fixed. Therefore, while the positions of \(A\) and \(B\) are constant, \(P\) coordinates vary, continuously shifting the centroid's position.
The centroid is always within the triangle and moves in a path that is dependent on the vertex movements. This path is what we seek to define as the locus of the centroid in the given problem.

Geometry of Circles

Circles are fundamental in geometry due to their unique properties and symmetry. Here, the circle in question has center \(O\) and radius \(r\). Several key aspects of circle geometry relate to this problem:

• The radius \(r\) is the distance from the center to any point on the circle, essential in calculating and understanding any scenarios involving circle properties and equations.
• The circle's symmetry indicates that movements around its circumference are consistent, which is useful for understanding locus pathways like in this exercise.
• Chords define internal angles and section the circle into measurable arcs and regions.

In our scenario, as point \(P\) moves, it maintains equal distance \(r\) from \(O\), and because \(AB\) always subtends this right angle, specific symmetric properties can be inferred about the resultant paths (locus) of points like the centroid \(G\). Understanding these principles is critical to analyzing how geometric configurations translate into specific curve shapes.

Right Angle Subtended by Chord

When a chord subtends a right angle at the center of a circle, it brings about certain geometrical advantages due to symmetry. In the context of this problem, chord \(AB\) subtending a right angle at \(O\) informs that \(AB\) divides the circle into two equal semicircles.
This setting implies that the segment \(AB\), being perpendicular to the radius at the midpoint, is actually a special configuration that leads to predictable symmetry in the locus of the centroid \(G\) of triangle \(PAB\). Because the configuration is symmetrical, when \(P\) moves along one semicircle, there's an inherent balance in the spatial distribution influencing \(G\)'s movement.
The subtended right angle implies a consistent property throughout the motion: any phrased motion or path described retains a particular relationship in distance and symmetry, suggesting that the locus of the centroid that we derive is indeed a circle. This is due to the circular symmetry revolving around called upon by the right angle subtended property.