A body is moving under the action of two force `vec(F_(1)=2hati-5hatj` , `vec(F_(2)=3hati-4hatj` . Its velocity will become uniform under a third force `vec(F_(3)` given by.

Options

A. `5hati-hatj`
B. `-5hati-hatj`
C. `5hati+hatj`
D. `-5hati+9hatj`

Explanation

To solve the problem, we need to find the third force \(\vec{F_3}\) that will make the net force acting on the body zero, thereby allowing the body to move with uniform velocity. This means that \(\vec{F_3}\) must be equal and opposite to the sum of the forces \(\vec{F_1}\) and \(\vec{F_2}\).

### Step-by-Step Solution:

1. **Identify the Given Forces:**
- \(\vec{F_1} = 2\hat{i} - 5\hat{j}\)
- \(\vec{F_2} = 3\hat{i} - 4\hat{j}\)

2. **Calculate the Resultant Force (\(\vec{F_R}\)):**
- The resultant force \(\vec{F_R}\) is the vector sum of \(\vec{F_1}\) and \(\vec{F_2}\).
\[ \vec{F_R} = \vec{F_1} + \vec{F_2} = (2\hat{i} - 5\hat{j}) + (3\hat{i} - 4\hat{j}) \]

3. **Combine the Components:**
- Combine the \(\hat{i}\) components and the \(\hat{j}\) components separately:
\[ \vec{F_R} = (2 + 3)\hat{i} + (-5 - 4)\hat{j} = 5\hat{i} - 9\hat{j} \]

4. **Determine the Third Force (\(\vec{F_3}\)):**
- Since we want the net force to be zero for uniform velocity, \(\vec{F_3}\) must be equal and opposite to \(\vec{F_R}\):
\[ \vec{F_3} = -\vec{F_R} = - (5\hat{i} - 9\hat{j}) = -5\hat{i} + 9\hat{j} \]

5. **Final Expression for \(\vec{F_3}\):**
- Therefore, the third force that will make the velocity uniform is:
\[ \vec{F_3} = -5\hat{i} + 9\hat{j} \]

### Conclusion:
The force \(\vec{F_3}\) that will make the velocity of the body uniform is \(-5\hat{i} + 9\hat{j}\).