Problem 71 - A Complete Resource Book in Mathematics for JEE Main 2017

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 answer is (B) \((3x - 1)^2 + (3y)^2 = a^2 + b^2\).

Step 1: Find the Centroid Coordinates

The centroid of a triangle with vertices at \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) is given by \(G(x, y) = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)\). Here, the vertices are \((a \cos t, a \sin t)\), \((b \sin t, -b \cos t)\), and \((1, 0)\). Calculating the coordinates of the centroid:\[ x = \frac{a \cos t + b \sin t + 1}{3}\]\[ y = \frac{a \sin t - b \cos t + 0}{3}\]

Step 2: Express in Terms of Known Parameters

We have the centroid coordinates as:\[ x = \frac{a \cos t + b \sin t + 1}{3}\]\[ y = \frac{a \sin t - b \cos t}{3}\]Multiply both equations by 3 to simplify:\[3x = a \cos t + b \sin t + 1\]\[3y = a \sin t - b \cos t\]

Step 3: Eliminate Trigonometric Terms

Square both the expressions:\[(3x - 1)^2 = (a \cos t + b \sin t)^2\]\[(3y)^2 = (a \sin t - b \cos t)^2\]Expand these squared terms:\[(3x - 1)^2 = a^2 \cos^2 t + 2ab \cos t \sin t + b^2 \sin^2 t\]\[(3y)^2 = a^2 \sin^2 t - 2ab \cos t \sin t + b^2 \cos^2 t\]

Step 4: Add the Equations

Adding the expanded equations:\[(3x - 1)^2 + (3y)^2 = \left(a^2 \cos^2 t + b^2 \sin^2 t + 2ab \cos t \sin t\right) + \left(a^2 \sin^2 t + b^2 \cos^2 t - 2ab \cos t \sin t\right)\]Combine and simplify the terms:\[(3x - 1)^2 + (3y)^2 = a^2 (\cos^2 t + \sin^2 t) + b^2 (\cos^2 t + \sin^2 t)\]Using \( \cos^2 t + \sin^2 t = 1 \), we get:\[(3x - 1)^2 + (3y)^2 = a^2 + b^2\]

Step 5: Determine the Correct Answer

The final equation \((3x - 1)^2 + (3y)^2 = a^2 + b^2\) represents the locus of the centroid. Comparing with the given options, this matches:Option (B): \((3x - 1)^2 + (3y)^2 = a^2 + b^2\).

Key Concepts

Centroid of Triangle

In geometry, the **centroid** of a triangle is an important concept and easy to understand with a simple formula. It refers to the point where the triangle's three medians intersect. Think of it as the triangle's center of balance. This centroid point is an average of the vertices' coordinates which, for a triangle with vertices

• \((x_1, y_1)\),
• \((x_2, y_2)\),
• \((x_3, y_3)\)

is expressed as:\[G(x, y) = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)\]Essentially, you find the average of the x-coordinates and the average of the y-coordinates to pinpoint this center. For any given triangle, the centroid provides useful insights into its geometry and can serve as a stepping stone for understanding more complex concepts.

Trigonometric Parameters

Trigonometry often comes into play when analyzing geometric figures like triangles. In this problem, the vertices of the triangle incorporate trigonometric expressions like \(a \cos t\) and \(b \sin t\). Understanding these **trigonometric parameters** is crucial for manipulating and simplifying expressions.The variables \(a\) and \(b\) act as coefficients that resize or scale the sine and cosine functions which are dependent on the angle \(t\). Trigonometric identities, such as \(\cos^2 t + \sin^2 t = 1\), are incredibly handy in solving complex expressions and are extensively used when simplifying these equations.Learn to recognize patterns and relationships between trigonometric expressions. Doing so will allow you to transform and simplify them effectively. When you grasp these parameters, you can tackle and solve intricate geometry problems with confidence.

Equation of Locus

In coordinate geometry, a **locus** is a set of points that satisfies a particular condition or rule. Understanding what a locus represents is pivotal in solving geometry problems. When working with a problem like finding the locus of a triangle's centroid, you are essentially looking for an equation that describes every possible position that the centroid could take.The original exercise aims to derive an equation representing the locus of the centroid of a triangle as its vertices move based on the parameter \(t\). After calculating and altering the expressions of the centroid's coordinates, the task was to arrive at an equation describing all possible centroids' positions, leading to the expression \((3x - 1)^2 + (3y)^2 = a^2 + b^2\).Once you conceptualize the idea of a locus, creating the equation becomes about identifying relations and constraints that govern the movement of points, like the centroid, in space.

Coordinate Geometry

**Coordinate geometry**, also known as analytic geometry, combines geometry and algebra and allows us to use coordinate points to represent geometric figures. It's especially powerful because it provides a way to calculate distances, midpoints, and other geometric properties using algebraic equations. In this context, coordinate geometry is utilized to compute the centroid of a triangle with given vertices and further explore its locus. By assigning coordinates to each vertex of the triangle, we compute the centroid and employ trigonometric identities to arrive at a single equation governing all its possible positions. Understanding coordinate geometry is crucial because it gives you the tools to analyze and solve geometric problems arithmetically. This approach can offer a clearer, more structured way to handle complex problems, proving to be an indispensable part of problem-solving in various fields of mathematics.