If ` vec b` is a vector whose initial point divides thejoin of `5 hat ia n d5 hat j` in the ratio `k :1` and whose terminal point is the origin and `| vec b|lt=sqrt(37),t h e nk` lies in the interval

Options

A. `[-6, -1//16]`
B. `(-oo, -6]uu[-1//6, oo)`
C. `[0, 6]`
D. none of these

Explanation

To solve the problem, we need to find the value of \( k \) given the conditions of the vector \( \vec{b} \). Let's go through the solution step by step.

### Step 1: Identify the Points
The initial point of vector \( \vec{b} \) divides the line segment joining the points \( 5 \hat{i} \) and \( 5 \hat{j} \) in the ratio \( k:1 \).

### Step 2: Find the Coordinates of the Dividing Point
Using the section formula, the coordinates of the point dividing the segment in the ratio \( k:1 \) can be calculated as:
\[ \text{Point} = \left( \frac{5 \cdot 1 + 5 \cdot k}{k + 1}, \frac{5 \cdot k + 5 \cdot 1}{k + 1} \right) = \left( \frac{5(1 + k)}{k + 1}, \frac{5(k + 1)}{k + 1} \right) = \left( 5, 5 \right) \]
This means the point is \( (5, 5) \).

### Step 3: Define the Vector \( \vec{b} \)
The vector \( \vec{b} \) has its initial point at \( (5, 5) \) and its terminal point at the origin \( (0, 0) \). Therefore, we can express \( \vec{b} \) as:
\[ \vec{b} = (0 - 5) \hat{i} + (0 - 5) \hat{j} = -5 \hat{i} - 5 \hat{j} \]

### Step 4: Calculate the Magnitude of \( \vec{b} \)
The magnitude of vector \( \vec{b} \) is given by:
\[ |\vec{b}| = \sqrt{(-5)^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2} \]

### Step 5: Set Up the Inequality
We are given that \( |\vec{b}| \leq \sqrt{37} \). Therefore, we need to satisfy the inequality:
\[ 5\sqrt{2} \leq \sqrt{37} \]

### Step 6: Square Both Sides
To eliminate the square roots, we square both sides:
\[ (5\sqrt{2})^2 \leq (\sqrt{37})^2 \]
This simplifies to:
\[ 50 \leq 37 \]
This inequality is false, indicating that our assumption about the magnitude must be checked against the conditions of \( k \).

### Step 7: Analyze the Condition
Since we need to find the values of \( k \) that satisfy the condition \( |\vec{b}| \leq \sqrt{37} \), we can derive that:
\[ \sqrt{25(1 + k^2)} \leq 37(k + 1) \]
Squaring both sides leads to:
\[ 25(1 + k^2) \leq 37(k^2 + 2k + 1) \]
Rearranging gives us:
\[ 0 \leq 12k^2 + 74k + 12 \]

### Step 8: Factor the Quadratic
Factoring or using the quadratic formula:
\[ k = \frac{-74 \pm \sqrt{74^2 - 4 \cdot 12 \cdot 12}}{2 \cdot 12} \]
This leads to the roots and intervals for \( k \).

### Step 9: Determine the Valid Range for \( k \)
By solving the quadratic inequality, we find that \( k \) lies in the intervals:
\[ k \in (-\infty, -6] \cup [-\frac{1}{6}, \infty) \]

### Conclusion
Thus, the final answer is that \( k \) lies in the interval \( (-\infty, -6) \cup (-\frac{1}{6}, \infty) \).