In a hydraulic lift, `F_(1) and F_(2)` are the force acting on the small piston and large piston having radii `r_(1) and r_(2)` rspectively, then

Options

A. `F_(1) = (r_(2)^(2))/(r_(1)^(2))F_(2)`
B. `F_(2) =(r_(2)^(2))/(r_(1)^(2))F_(1)`
C. `F_(1)=(r_(1))/(r_(2))F_(2)`
D. `F_(1) =(r_(2))/(r_(1))F_(2)`

Answer

To solve the problem regarding the relationship between the forces \( F_1 \) and \( F_2 \) acting on the small and large pistons of a hydraulic lift, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Concept**:
In a hydraulic lift, the pressure exerted on both pistons is the same. This is due to Pascal's principle which states that pressure applied to a confined fluid is transmitted undiminished in all directions.

2. **Define the Variables**:
- Let \( F_1 \) be the force acting on the small piston.
- Let \( F_2 \) be the force acting on the large piston.
- Let \( r_1 \) be the radius of the small piston.
- Let \( r_2 \) be the radius of the large piston.

3. **Calculate the Areas**:
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
Therefore, the areas of the small and large pistons can be expressed as:
- Area of small piston, \( A_1 = \pi r_1^2 \)
- Area of large piston, \( A_2 = \pi r_2^2 \)

4. **Set Up the Pressure Equation**:
Since the pressure is the same on both sides, we can write:
\[ P_1 = P_2 \]
where \( P_1 \) and \( P_2 \) are the pressures on the small and large pistons, respectively. Pressure is defined as force per unit area:
\[ P_1 = \frac{F_1}{A_1} \quad \text{and} \quad P_2 = \frac{F_2}{A_2} \]

5. **Equate the Pressures**:
Substituting the areas into the pressure equation gives:
\[ \frac{F_1}{\pi r_1^2} = \frac{F_2}{\pi r_2^2} \]

6. **Simplify the Equation**:
The \( \pi \) cancels out from both sides:
\[ \frac{F_1}{r_1^2} = \frac{F_2}{r_2^2} \]

7. **Rearranging for \( F_1 \) and \( F_2 \)**:
Rearranging the equation to express \( F_1 \) in terms of \( F_2 \):
\[ F_1 = F_2 \cdot \frac{r_1^2}{r_2^2} \]
Alternatively, expressing \( F_2 \) in terms of \( F_1 \):
\[ F_2 = F_1 \cdot \frac{r_2^2}{r_1^2} \]

8. **Conclusion**:
The relationship between the forces \( F_1 \) and \( F_2 \) can be summarized as:
\[ F_2 = F_1 \cdot \frac{r_2^2}{r_1^2} \]

### Final Answer:
Thus, the correct option that represents the relationship between \( F_1 \) and \( F_2 \) is:
\[ F_2 = F_1 \cdot \frac{r_2^2}{r_1^2} \]