If a line has a vector equation, `vecr=2hati +6hatj+lambda(hati-3hatj)` then which of the following statements holds good ?

Options

A. the line is parallel to `2hati +6hatj`
B. the line passes through the point `3hati+3hatj`
C. the line passes through the point `hati+9hatj`
D. the line is parallel to xy plane

Answer

To solve the problem, we need to analyze the given vector equation of the line and determine which statements are correct based on that equation.

### Given Vector Equation:
The vector equation of the line is given as:
\[ \vec{r} = 2\hat{i} + 6\hat{j} + \lambda(\hat{i} - 3\hat{j}) \]

### Step 1: Identify the Point and Direction Vector
From the vector equation, we can identify:
- The point through which the line passes: \( P(2, 6, 0) \)
- The direction vector of the line: \( \vec{d} = \hat{i} - 3\hat{j} \)

### Step 2: Analyze the Statements
Now, we will analyze each of the statements provided in the options.

#### Option A: The line is parallel to \( 2\hat{i} + 6\hat{j} \)
To check if the line is parallel to this vector, we compare the direction vector of the line \( \vec{d} = \hat{i} - 3\hat{j} \) with \( 2\hat{i} + 6\hat{j} \).

1. The direction vector of the line is \( \hat{i} - 3\hat{j} \).
2. For two vectors to be parallel, they must be scalar multiples of each other.

Setting:
\[ \hat{i} - 3\hat{j} = k(2\hat{i} + 6\hat{j}) \]

By comparing coefficients:
- For \( \hat{i} \): \( 1 = 2k \) → \( k = \frac{1}{2} \)
- For \( \hat{j} \): \( -3 = 6k \) → \( k = -\frac{1}{2} \)

Since we have a contradiction in the value of \( k \), the line is **not parallel** to \( 2\hat{i} + 6\hat{j} \). Thus, **Option A is incorrect**.

#### Option B: The line passes through \( 3\hat{i} + 3\hat{j} \)
To check if the line passes through the point \( (3, 3, 0) \):

1. Set \( \vec{r} = 3\hat{i} + 3\hat{j} \).
2. From the vector equation:
\[ 3\hat{i} + 3\hat{j} = 2\hat{i} + 6\hat{j} + \lambda(\hat{i} - 3\hat{j}) \]
This leads to the equations:
- \( 2 + \lambda = 3 \) → \( \lambda = 1 \)
- \( 6 - 3\lambda = 3 \) → \( 6 - 3(1) = 3 \) → True.

Both equations are satisfied, so **Option B is correct**.

#### Option C: The line passes through \( \hat{i} + 9\hat{j} \)
To check if the line passes through the point \( (1, 9, 0) \):

1. Set \( \vec{r} = \hat{i} + 9\hat{j} \).
2. From the vector equation:
\[ \hat{i} + 9\hat{j} = 2\hat{i} + 6\hat{j} + \lambda(\hat{i} - 3\hat{j}) \]
This leads to the equations:
- \( 2 + \lambda = 1 \) → \( \lambda = -1 \)
- \( 6 - 3\lambda = 9 \) → \( 6 - 3(-1) = 9 \) → True.

Both equations are satisfied, so **Option C is correct**.

#### Option D: The line is parallel to the XY Plane
A line is parallel to the XY plane if its direction vector has no component in the Z direction. Since our direction vector \( \hat{i} - 3\hat{j} \) has no \( \hat{k} \) component, the line is indeed parallel to the XY plane. Thus, **Option D is correct**.

### Conclusion
The correct options are:
- **Option B**: The line passes through \( 3\hat{i} + 3\hat{j} \).
- **Option C**: The line passes through \( \hat{i} + 9\hat{j} \).
- **Option D**: The line is parallel to the XY Plane.