The vertices of triangle are A(-5,3),B(p-1) and C(6,q). If the centroid of the `DeltaABC` is `(1,-1)`, then the values of p and q are
(i) `p=-2,q=5`
(ii) `p=2,q=-5`
(iii) `p=3,q=5`
(iv) `p=-5,q=2`

Options

A. `p=-2,q=5`
B. `p=2,q=-5`
C. `p=3,q=5`
D. `p=-5,q=2`

Answer

To find the values of \( p \) and \( q \) for the vertices of triangle \( A(-5, 3) \), \( B(p, -1) \), and \( C(6, q) \) given that the centroid is \( (1, -1) \), we can use the formula for the centroid of a triangle.

### Step-by-step Solution:

1. **Understanding the Centroid Formula**:
The coordinates of the centroid \( G \) of a triangle with vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) are given by:
\[ G_x = \frac{x_1 + x_2 + x_3}{3}, \quad G_y = \frac{y_1 + y_2 + y_3}{3} \]

2. **Substituting the Known Values**:
For our triangle:
- \( A(-5, 3) \) gives \( x_1 = -5 \) and \( y_1 = 3 \)
- \( B(p, -1) \) gives \( x_2 = p \) and \( y_2 = -1 \)
- \( C(6, q) \) gives \( x_3 = 6 \) and \( y_3 = q \)

The centroid is given as \( (1, -1) \).

3. **Setting Up the Equations**:
From the x-coordinate of the centroid:
\[ \frac{-5 + p + 6}{3} = 1 \]
Simplifying this:
\[ \frac{p + 1}{3} = 1 \]
Multiplying both sides by 3:
\[ p + 1 = 3 \]
Thus,
\[ p = 2 \]

4. **Finding the y-coordinate**:
From the y-coordinate of the centroid:
\[ \frac{3 - 1 + q}{3} = -1 \]
Simplifying this:
\[ \frac{2 + q}{3} = -1 \]
Multiplying both sides by 3:
\[ 2 + q = -3 \]
Thus,
\[ q = -3 - 2 = -5 \]

5. **Final Values**:
Therefore, the values of \( p \) and \( q \) are:
\[ p = 2, \quad q = -5 \]

### Conclusion:
The correct answer is \( (p, q) = (2, -5) \), which corresponds to option (ii).