The vertices of a triangle are `(1, a), (2, b) and (c^2, -3).` (i) Prove that its centroid can not lie on the y-axis. (ii) Find the condition that the centroid may lie on the x-axis for any value of `a, b, c in RR.`

Answer

To solve the given problem step by step, we will analyze the vertices of the triangle and calculate the centroid.

### Step 1: Identify the vertices of the triangle
The vertices of the triangle are given as:
- \( A(1, a) \)
- \( B(2, b) \)
- \( C(c^2, -3) \)

### Step 2: Calculate the coordinates of the centroid
The formula for the centroid \( G \) of a triangle with vertices \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \) is given by:
\[ G\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]

Substituting the coordinates of the vertices:
\[ G\left(\frac{1 + 2 + c^2}{3}, \frac{a + b - 3}{3}\right) = G\left(\frac{3 + c^2}{3}, \frac{a + b - 3}{3}\right) \]

### Step 3: Prove that the centroid cannot lie on the y-axis
For the centroid to lie on the y-axis, its x-coordinate must be zero:
\[ \frac{3 + c^2}{3} = 0 \]

Multiplying both sides by 3 gives:
\[ 3 + c^2 = 0 \]

Rearranging gives:
\[ c^2 = -3 \]

Since the square of any real number cannot be negative, \( c^2 = -3 \) has no real solutions. Therefore, we conclude:
\[ \text{The centroid cannot lie on the y-axis.} \]

### Step 4: Find the condition for the centroid to lie on the x-axis
For the centroid to lie on the x-axis, its y-coordinate must be zero:
\[ \frac{a + b - 3}{3} = 0 \]

Multiplying both sides by 3 gives:
\[ a + b - 3 = 0 \]

Rearranging gives:
\[ a + b = 3 \]

### Conclusion
1. The centroid cannot lie on the y-axis.
2. The condition for the centroid to lie on the x-axis is \( a + b = 3 \).