If ` vec b` is a vector whose initial point divides the join of `5 hat i and 5 hat j` in the ratio `k :1` and whose terminal point is the origin and `| vec b|lt=sqrt(37),then , k` lies in the interval
a. `[-6,-1//6]`
b. `(-oo,-6]uu[-1//6,oo)`
c. `[0,6]`
d. none of these

Explanation

To solve the problem step by step, we will use the section formula and properties of vectors.

### Step 1: Identify the points and the ratio
The initial point divides the line segment joining the points \(5\hat{i}\) and \(5\hat{j}\) in the ratio \(k:1\). Therefore, we can denote the points as:
- Point A: \(5\hat{i}\)
- Point B: \(5\hat{j}\)

### Step 2: Apply the section formula
Using the section formula, the coordinates of the point that divides the line segment joining \(A\) and \(B\) in the ratio \(k:1\) is given by:
\[ \vec{P} = \frac{k \cdot \vec{B} + 1 \cdot \vec{A}}{k + 1} \]
Substituting \(\vec{A} = 5\hat{i}\) and \(\vec{B} = 5\hat{j}\):
\[ \vec{P} = \frac{k \cdot (5\hat{j}) + 1 \cdot (5\hat{i})}{k + 1} = \frac{5k\hat{j} + 5\hat{i}}{k + 1} \]
This simplifies to:
\[ \vec{P} = \frac{5\hat{i} + 5k\hat{j}}{k + 1} \]

### Step 3: Determine the vector \(\vec{b}\)
The vector \(\vec{b}\) has its initial point at \(\vec{P}\) and its terminal point at the origin (0,0). Therefore:
\[ \vec{b} = \vec{P} - \vec{0} = \frac{5\hat{i} + 5k\hat{j}}{k + 1} \]

### Step 4: Calculate the magnitude of \(\vec{b}\)
The magnitude of \(\vec{b}\) is given by:
\[ |\vec{b}| = \left| \frac{5\hat{i} + 5k\hat{j}}{k + 1} \right| = \frac{1}{k + 1} \sqrt{(5)^2 + (5k)^2} = \frac{1}{k + 1} \sqrt{25 + 25k^2} = \frac{5\sqrt{1 + k^2}}{k + 1} \]

### Step 5: Set up the inequality
According to the problem, we have:
\[ |\vec{b}| < \sqrt{37} \]
This leads to the inequality:
\[ \frac{5\sqrt{1 + k^2}}{k + 1} < \sqrt{37} \]

### Step 6: Square both sides
Squaring both sides gives:
\[ \frac{25(1 + k^2)}{(k + 1)^2} < 37 \]
Cross-multiplying leads to:
\[ 25(1 + k^2) < 37(k + 1)^2 \]

### Step 7: Expand and simplify
Expanding both sides:
\[ 25 + 25k^2 < 37(k^2 + 2k + 1) \]
This simplifies to:
\[ 25 + 25k^2 < 37k^2 + 74k + 37 \]
Rearranging gives:
\[ 0 < 12k^2 + 74k + 12 \]

### Step 8: Solve the quadratic inequality
To find the values of \(k\) for which this inequality holds, we need to find the roots of the quadratic equation:
\[ 12k^2 + 74k + 12 = 0 \]
Using the quadratic formula:
\[ k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-74 \pm \sqrt{74^2 - 4 \cdot 12 \cdot 12}}{2 \cdot 12} \]
Calculating the discriminant:
\[ = \frac{-74 \pm \sqrt{5476 - 576}}{24} = \frac{-74 \pm \sqrt{4900}}{24} = \frac{-74 \pm 70}{24} \]
This gives the roots:
\[ k_1 = \frac{-4}{24} = -\frac{1}{6}, \quad k_2 = \frac{-144}{24} = -6 \]

### Step 9: Determine the intervals
The quadratic \(12k^2 + 74k + 12\) opens upwards (as the coefficient of \(k^2\) is positive). Thus, the inequality \(12k^2 + 74k + 12 > 0\) holds for:
\[ k \in (-\infty, -6) \cup \left(-\frac{1}{6}, \infty\right) \]

### Conclusion
Thus, the correct answer is option **b**: \((-∞, -6] \cup [-\frac{1}{6}, ∞)\).