To solve the problem, we need to calculate the work done by the force \(\vec{F} = y \hat{i} + x \hat{j}\) when a particle moves from point A (1 m, 1 m) to point B (9 m, 3 m) along a straight path. The work done is given as \(y\).
### Step-by-Step Solution:
1. **Identify the Force and Path:**
The force acting on the particle is given by:
\[
\vec{F} = y \hat{i} + x \hat{j}
\]
The initial position (point A) is \((x_1, y_1) = (1, 1)\) and the final position (point B) is \((x_2, y_2) = (9, 3)\).
2. **Displacement Vector:**
The displacement vector \(d\vec{r}\) can be expressed as:
\[
d\vec{r} = dx \hat{i} + dy \hat{j}
\]
3. **Work Done Calculation:**
The work done \(W\) by the force when moving from point A to point B is given by the line integral:
\[
W = \int_{A}^{B} \vec{F} \cdot d\vec{r}
\]
Substituting the expressions for \(\vec{F}\) and \(d\vec{r}\):
\[
W = \int_{A}^{B} (y \hat{i} + x \hat{j}) \cdot (dx \hat{i} + dy \hat{j}) = \int_{A}^{B} (y \, dx + x \, dy)
\]
4. **Setting Up the Limits:**
We need to integrate with respect to \(x\) and \(y\). The limits for \(x\) are from 1 to 9, and for \(y\) from 1 to 3.
5. **Calculate the Work Done:**
We can break the integral into two parts:
\[
W = \int_{1}^{9} y \, dx + \int_{1}^{3} x \, dy
\]
- For the first integral, since \(y\) is changing from 1 to 3, we can express \(y\) in terms of \(x\) using the straight line equation connecting points A and B. The slope of the line is:
\[
\text{slope} = \frac{3 - 1}{9 - 1} = \frac{2}{8} = \frac{1}{4}
\]
Thus, the equation of the line is:
\[
y - 1 = \frac{1}{4}(x - 1) \implies y = \frac{1}{4}x + \frac{3}{4}
\]
- Now substituting \(y\) into the first integral:
\[
W_1 = \int_{1}^{9} \left(\frac{1}{4}x + \frac{3}{4}\right) dx
\]
- For the second integral, we can substitute \(x\) in terms of \(y\) using the same line equation:
\[
x = 4y - 1
\]
Thus:
\[
W_2 = \int_{1}^{3} (4y - 1) dy
\]
6. **Evaluating the Integrals:**
- For \(W_1\):
\[
W_1 = \left[\frac{1}{8}x^2 + \frac{3}{4}x\right]_{1}^{9} = \left[\frac{1}{8}(81) + \frac{27}{4}\right] - \left[\frac{1}{8}(1) + \frac{3}{4}\right]
\]
- For \(W_2\):
\[
W_2 = \left[2y^2 - y\right]_{1}^{3} = \left[2(9) - 3\right] - \left[2(1) - 1\right]
\]
7. **Final Calculation:**
After calculating both integrals, we combine them to find the total work done \(W\) and set it equal to \(y\). From this, we can solve for the ratio \(\frac{x}{y}\).
8. **Finding the Ratio:**
After simplifying the equations, we find:
\[
7y + 2x = 0 \implies \frac{x}{y} = -\frac{7}{2}
\]
### Final Answer:
The value of \(\frac{x}{y}\) is \(-\frac{7}{2}\).
