Problem 71 - A Complete Resource Book in Mathematics for JEE Main

Question

Locus of centroid of the triangle whose vertices are (a \(\cos t, a \sin t),(b \sin t,-b \cos t)\) and \((1,0)\), where \(t\) is a parameter, is (A) \((3 x-1)^{2}+(3 y)^{2}=a^{2}-b^{2}\) (B) \((3 x-1)^{2}+(3 y)^{2}=a^{2}+b^{2}\) (C) \((3 x+1)^{2}+(3 y)^{2}=a^{2}+b^{2}\) (D) \((3 x+1)^{2}+(3 y)^{2}=a^{2}-b^{2}\)

Step-by-Step Solution

Answer

The locus of the centroid is given by option (B): \((3x - 1)^2 + (3y)^2 = a^2 + b^2\).

Step 1: Calculate the Centroid Coordinates

To find the coordinates of the centroid of the triangle with vertices \((x_1, y_1), (x_2, y_2), (x_3, y_3)\), use the formula: \( \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \). For this triangle, the vertices are \((a \cos t, a \sin t)\), \((b \sin t, -b \cos t)\), and \((1, 0)\). So the centroid \((x, y)\) is given by: \[ x = \frac{a \cos t + b \sin t + 1}{3}, \quad y = \frac{a \sin t - b \cos t + 0}{3} \] \[ x = \frac{a \cos t + b \sin t + 1}{3}, \quad y = \frac{a \sin t - b \cos t}{3} \].

Step 2: Express the Centroid Locus Equation

The goal is to relate \(x\) and \(y\) without the parameter \(t\). Start with both expressions: \(3x = a \cos t + b \sin t + 1\) and \(3y = a \sin t - b \cos t\). Square both sides and add them together: \[ (3x - 1)^2 = (a \cos t + b \sin t)^2 \] \[ (3y)^2 = (a \sin t - b \cos t)^2 \] \[ (3x - 1)^2 + (3y)^2 = a^2 \cos^2 t + 2ab \cos t \sin t + b^2 \sin^2 t + a^2 \sin^2 t - 2ab \sin t \cos t + b^2 \cos^2 t \].

Step 3: Simplify the Expression

Combine like terms in the equation from the previous step: \((3x - 1)^2 + (3y)^2 = a^2(\cos^2 t + \sin^2 t) + b^2(\sin^2 t + \cos^2 t) \)). Since \(\cos^2 t + \sin^2 t = 1\), the equation becomes: \[ (3x - 1)^2 + (3y)^2 = a^2 + b^2 \].

Step 4: Conclusion of the Locus Problem

The resultant equation \((3x - 1)^2 + (3y)^2 = a^2 + b^2\) matches one of the given options. Therefore, option (B) is the correct answer.

Key Concepts

Centroid of a Triangle

The centroid of a triangle is an important concept in geometry. It is the point where the three medians of the triangle intersect. To find the centroid's coordinates in a triangle with vertices at

• (x₁, y₁),
• (x₂, y₂),
• (x₃, y₃),

you simply average each set of coordinates. The formula is: \[ \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \] This formula provides the centroid's x and y coordinates by taking the mean of the x-coordinates and the mean of the y-coordinates of all the vertices. It's a straightforward calculation! For a triangle with vertices \((a \cos t, a \sin t)\), \((b \sin t, -b \cos t)\), and \((1, 0)\), plug these values into the formula to find the centroid. Understanding the centroid is crucial for determining how the triangle balances because it's often thought of as the triangle's center of mass. This concept not only applies to coordinate geometry but also has practical implications in physics and engineering.

Locus Equation

The locus equation describes a set of points that satisfy a particular condition or relation in geometry. In this exercise, we want to find the locus of the centroid as one of the vertices of the triangle changes with a parameter \(t\). To derive the locus equation, we find an expression for the centroid's coordinates and then eliminate the parameter \(t\). For a triangle with parametric vertices such as \((a \cos t, a \sin t)\) and \((b \sin t, -b \cos t)\), we seek to express the relation between \(x\) and \(y\) that holds for all values of \(t\). To achieve this, use trigonometric identities and algebraic manipulation, as shown in the solution:

• Square both expressions for \(x\) and \(y\) in terms of \(t\).
• Add the squared equations together.
• Simplify using identities, such as \(\cos^2 t + \sin^2 t = 1\).

The outcome is the equation \((3x - 1)^2 + (3y)^2 = a^2 + b^2\), capturing the locus of all possible centroids for the triangle as \(t\) varies. This process of finding the locus is beneficial in various mathematical and real-world applications, as it provides a way to describe the behavior and constraint of a dynamic point geometrically.

Parametric Equations

Parametric equations are a way to express a set of related mathematical quantities as explicit functions of a variable, known as a parameter. Unlike standard form equations, which relate \(x\) and \(y\) directly, parametric equations define both \(x\) and \(y\) in terms of another variable \(t\). For example, the points \((x_1, y_1) = (a \cos t, a \sin t)\) are given in terms of \(t\), where \(a\) is a constant radius of a circle. This approach is particularly useful for describing curves, circles, and motion in two-dimensional space.

• It allows for a concise representation of complex shapes.
• It facilitates the analysis of motion, as time often serves as \(t\).
• Disentangle geometrical constraints more easily during calculus operations.

In this exercise, understanding how parametric equations define triangle vertices helps deducing how the centroid changes its position as the parameter \(t\) varies. The relationship provides a unique view on how elements move and interact in coordinate geometry, allowing deeper insights into paths and transformations.