If `v_(1)` and `v_(2)` be the velocities at the end of focal chord of projectile path and u is the velocity at the vertex of the path, then

Options

A. `(1)/(v^(1))+(1)/(v^(2))=(1)/(u)`
B. `(1)/(v_(2)^(2))+(1)/(v_(2)^(2))=(1)/(u^(2))`
C. `v_(1)^(2)+v_(2)^(2)=u^(2)`
D. `u=v_(1)+v_(2)`

Answer

To solve the problem, we need to analyze the relationship between the velocities at the ends of the focal chord of a projectile path and the velocity at the vertex of the path.

### Step-by-Step Solution:

1. **Understanding the Focal Chord**:
- A focal chord is a line segment that passes through the focus of a conic section (in this case, a parabola) and intersects it at two points. The velocities at these points are denoted as \( v_1 \) and \( v_2 \).

2. **Velocity at the Vertex**:
- The vertex of the projectile path is the highest point in its trajectory, where the velocity is denoted as \( u \).

3. **Using the Given Relationship**:
- The problem states that:
\[ \frac{1}{v_1} + \frac{1}{v_2} = \frac{1}{u} \]
- This equation indicates a relationship between the velocities at the ends of the focal chord and the velocity at the vertex.

4. **Rearranging the Equation**:
- We can rearrange the equation to express it in a different form:
\[ \frac{1}{u} = \frac{1}{v_1} + \frac{1}{v_2} \]
- This implies that the harmonic mean of \( v_1 \) and \( v_2 \) is equal to \( u \).

5. **Verifying Other Options**:
- **Option B**:
\[ \frac{1}{v_1^2} + \frac{1}{v_2^2} = \frac{1}{u^2} \]
- This is not generally true based on the harmonic mean relationship.

- **Option C**:
\[ v_1^2 + v_2^2 = u^2 \]
- This is also not a valid relationship derived from the given information.

- **Option D**:
\[ u = v_1 + v_2 \]
- This is not correct as it does not follow from the initial equation.

6. **Conclusion**:
- The only correct option based on the analysis is:
\[ \frac{1}{v_1} + \frac{1}{v_2} = \frac{1}{u} \]
- Therefore, option A is correct.