Two balloons are moving in air with velocities `v_(1)={2thati+(t-2)hatj)m//s` and `v_(2)={(t-4)hati+thatj}m//s` then at what `t` balloons are moving parallel to each other:

Options

A. `5//4s`
B. `4//5s`
C. `10//3s`
D. `-3+sqrt(17)s`

Answer

To find the time \( t \) at which the two balloons are moving parallel to each other, we need to analyze their velocity vectors and set up a condition for parallelism.

### Step-by-Step Solution:

1. **Identify the velocity vectors:**
The velocities of the two balloons are given as:
\[ \mathbf{v_1} = 2t \hat{i} + (t - 2) \hat{j} \quad \text{(1)} \]
\[ \mathbf{v_2} = (t - 4) \hat{i} + t \hat{j} \quad \text{(2)} \]

2. **Condition for parallel vectors:**
Two vectors \( \mathbf{a} \) and \( \mathbf{b} \) are parallel if the ratios of their corresponding components are equal:
\[ \frac{a_x}{b_x} = \frac{a_y}{b_y} \]
Applying this to our velocity vectors, we have:
\[ \frac{2t}{t - 4} = \frac{t - 2}{t} \quad \text{(3)} \]

3. **Cross-multiplying to eliminate the fractions:**
Cross-multiplying equation (3) gives:
\[ 2t^2 = (t - 2)(t - 4) \]

4. **Expanding the right-hand side:**
Expanding the right-hand side:
\[ 2t^2 = t^2 - 4t - 2t + 8 \]
Simplifying:
\[ 2t^2 = t^2 - 6t + 8 \]

5. **Rearranging to form a quadratic equation:**
Rearranging gives:
\[ 2t^2 - t^2 + 6t - 8 = 0 \]
Which simplifies to:
\[ t^2 + 6t - 8 = 0 \quad \text{(4)} \]

6. **Using the quadratic formula to solve for \( t \):**
The quadratic formula is given by:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = 6 \), and \( c = -8 \). Plugging these values into the formula:
\[ t = \frac{-6 \pm \sqrt{6^2 - 4 \cdot 1 \cdot (-8)}}{2 \cdot 1} \]
\[ t = \frac{-6 \pm \sqrt{36 + 32}}{2} \]
\[ t = \frac{-6 \pm \sqrt{68}}{2} \]
\[ t = \frac{-6 \pm 2\sqrt{17}}{2} \]
\[ t = -3 \pm \sqrt{17} \quad \text{(5)} \]

7. **Selecting the valid solution:**
Since time cannot be negative, we discard the solution \( t = -3 - \sqrt{17} \). Thus, we have:
\[ t = -3 + \sqrt{17} \quad \text{(6)} \]

### Final Answer:
The time at which the balloons are moving parallel to each other is:
\[ t = -3 + \sqrt{17} \text{ seconds} \]