Key Concepts and Formulas

• Centroid of a Triangle: The centroid of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is given by $\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right)$.
• Trigonometric Identity: $\sin^2 t + \cos^2 t = 1$.
• Locus: The locus of a point is the path traced by the point as it moves according to a given condition. To find the locus, we express the coordinates of the point in terms of a parameter and then eliminate the parameter to obtain an equation relating the coordinates.

Step-by-Step Solution

Step 1: Define the vertices and the centroid.

Let the vertices of the triangle be $P_1(a\cos t, a\sin t)$, $P_2(b\sin t, -b\cos t)$, and $P_3(1, 0)$. Let the centroid of the triangle be $(x, y)$. Our goal is to find the equation relating $x$ and $y$, which represents the locus of the centroid.

Step 2: Apply the centroid formula.

Using the centroid formula, we have: $x = \frac{a\cos t + b\sin t + 1}{3}$ $y = \frac{a\sin t - b\cos t + 0}{3}$

Step 3: Rearrange the equations.

Rearrange the equations to isolate the terms involving $t$: $3x - 1 = a\cos t + b\sin t \quad \text{(Equation 1)}$ $3y = a\sin t - b\cos t \quad \text{(Equation 2)}$

Step 4: Eliminate the parameter 't'.

Square both sides of Equation 1 and Equation 2: $(3x - 1)^2 = (a\cos t + b\sin t)^2 = a^2\cos^2 t + 2ab\cos t\sin t + b^2\sin^2 t \quad \text{(Equation 3)}$ $(3y)^2 = (a\sin t - b\cos t)^2 = a^2\sin^2 t - 2ab\sin t\cos t + b^2\cos^2 t \quad \text{(Equation 4)}$

Add Equation 3 and Equation 4: $(3x - 1)^2 + (3y)^2 = a^2\cos^2 t + 2ab\cos t\sin t + b^2\sin^2 t + a^2\sin^2 t - 2ab\sin t\cos t + b^2\cos^2 t$ $(3x - 1)^2 + (3y)^2 = a^2(\cos^2 t + \sin^2 t) + b^2(\sin^2 t + \cos^2 t)$

Step 5: Simplify using the trigonometric identity.

Using the identity $\sin^2 t + \cos^2 t = 1$, we get: $(3x - 1)^2 + (3y)^2 = a^2(1) + b^2(1)$ $(3x - 1)^2 + (3y)^2 = a^2 + b^2$

Step 6: State the locus equation.

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

Common Mistakes & Tips:

• Carefully apply the centroid formula, ensuring you add all x-coordinates and y-coordinates correctly before dividing by 3.
• When squaring binomials, remember to include the cross term (e.g., $(a+b)^2 = a^2 + 2ab + b^2$).
• Make sure to use the trigonometric identity $\sin^2 t + \cos^2 t = 1$ to simplify and eliminate the parameter $t$.

Summary:

We found the locus of the centroid of the triangle by first expressing the coordinates of the centroid in terms of the parameter $t$. Then, we eliminated $t$ by squaring and adding the equations, using the trigonometric identity $\sin^2 t + \cos^2 t = 1$. The resulting equation represents the locus of the centroid: $(3x - 1)^2 + (3y)^2 = a^2 + b^2$, which corresponds to option (C).

Final Answer

The final answer is $\boxed{(3x - 1)^2 + (3y)^2 = a^2 + b^2}$, which corresponds to option (C).