Statement-I `(S_1)`: If `A(x_1, y_1), B(x_2, y_2), C(x_3, y_3)` are non-collinear points. Then, every point (x, y) in the plane of `triangleABC`, can be expressed in the form `((kx_1+lx_2+mx_3)/(k+l+m), (ky_1+ly_2+my_3)/(k+l+m))`
Statement-II `(S_2)` The condition for coplanarity of four A(a), B(b), C(c), D(d) is that there exists scalars l, m, n, p not all zeros such that `la+mb+nc+pd=0` where `l+m+n+p=0`.

Options

A. Both Statement-I and Statement-II are correct and Statement-II is the correct explanation of Statement-I
B. Both Statement-I and Statement-II are correct but Statement-II is not the correct explanation of Statement-I
C. Statement-I is correct but Statement-II is incorrect
D. Statement-II is correct but Statement-I is incorrect

Answer

To solve the problem, we will analyze both statements and provide a step-by-step explanation for each.

### Step 1: Understanding Statement I (S1)
Statement I states that if points A(x1, y1), B(x2, y2), and C(x3, y3) are non-collinear, then any point (x, y) in the plane of triangle ABC can be expressed in a specific form involving weights k, l, and m.

**Mathematical Formulation:**
The point (x, y) can be expressed as:
\[ x = \frac{kx_1 + lx_2 + mx_3}{k + l + m} \]
\[ y = \frac{ky_1 + ly_2 + my_3}{k + l + m} \]

### Step 2: Proving Statement I
To prove this, consider the following:

1. **Non-collinearity**: Since A, B, and C are non-collinear, they form a triangle, and any point P in the plane can be represented as a linear combination of the vertices A, B, and C.

2. **Weights**: The weights k, l, and m represent how much of each vertex contributes to the point P. The sum \( k + l + m \) normalizes the weights.

Thus, any point (x, y) in the triangle can indeed be represented in this form, confirming that Statement I is true.

### Step 3: Understanding Statement II (S2)
Statement II states that for four points A(a), B(b), C(c), and D(d) to be coplanar, there must exist scalars l, m, n, and p (not all zero) such that:
\[ la + mb + nc + pd = 0 \]
with the condition:
\[ l + m + n + p = 0 \]

### Step 4: Proving Statement II
To prove this, we can use the following reasoning:

1. **Coplanarity Condition**: The condition for coplanarity of four points in vector form is that the vectors formed by these points can be expressed as a linear combination of each other.

2. **Linear Combination**: If we express the vector D as a linear combination of the vectors formed by A, B, and C, we can write:
\[ D = \lambda A + \mu B + \nu C \]
for some scalars λ, μ, and ν.

3. **Setting Up the Equation**: Rearranging gives us:
\[ D - \lambda A - \mu B - \nu C = 0 \]
This can be rewritten in the form of the condition given in Statement II.

4. **Verifying the Sum**: The condition \( l + m + n + p = 0 \) ensures that the vectors are indeed coplanar.

Thus, Statement II is also true.

### Conclusion
Both statements are true, and Statement II provides a correct explanation for Statement I. Therefore, the correct answer is:
**Both statements are correct, and Statement II is the correct explanation for Statement I.**